Physics Book - User contributions [en]

Electromagnetic Radiation

2023-12-08T22:18:31Z

Daniel72: Symbol is lambda, not gamma. /* General Properties */

'''Claimed by Zoila de Leon''' (Fall 2023)
==Electromagnetic Radiation==

===What is a Electromagnetic(EM) Radiation?===
Electromagnetic radiation is a form of energy that is all around us and takes many forms, such as radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma rays.

Before 1873, electricity and magnetism were thought to be two different forces. However, in 1873, Scottish Physicist James Maxwell developed his famous theory of electromagnetism. There are four main electro magnetic interactions according to Maxwell:
* The force of attraction or repulsion between electric charges is inversely proportional to the square of the distance between them
* Magnetic poles come in pairs that attract and repel each other much as electric charges do
* An electric current in a wire produces a magnetic field whose direction depends on the direction of the current
* A moving electric field produces a magnetic field, and vice versa

===General Properties===
The four Maxwell's Equations provide a complete description of possible spatial patterns of electric and magnetic field in space.

<div class="mw-collapsible-content">
*[[The Ampere-Maxwell Law]]
<div class="mw-collapsible-content">
*[[Gauss's Law]]
<div class="mw-collapsible-content">
*[[Faraday's Law]]

Other than Maxwell's Four equations, there are general properties of all electromagnetic radiation:
* Electromagnetic radiation can travel through empty space. Most other types of waves must travel through some sort of substance. For example, sound waves need either a gas, solid, or liquid to pass through in order to be heard
* The speed of light is always a constant (3 x 10^8 m/s)
* Wavelengths are measured between the distances of either crests or troughs. It is usually characterized by the Greek symbol λ (lambda).

Electromagnetic waves are the self-propagating, mutual oscillation of electric and magnetic fields. The propagation of electromagnetic energy is often referred to as radiation. We can also say that the 'pulse' of these moving fields result in radiation (7).

The equation for propagation is E=cB with c being the speed of light. This equation is derived from combining the two equations E=vB and B=u0e0vE, proving that v is equal to 3e8 meters/second.

===Problem Solving Method and Equations===

To go about solving/analyzing mathematically an electromagnetic field using Maxwell's equations,this is how we proceed (7)

*Establish the space and time in which the electric and magnetic fields are present
*Check that Maxwell's equations can be applied in the situation above
*Check when the charge accelerates, it produces these fields and therefore radiation
*Show how these fields would interact with matter

The equation of the Radiative Electric Field is:
E= 1/(4πe0)*-qa/(c^2r) where a is the acceleration of the particle, c is the speed of light and r is the distance from the original location of the charge to right before the kink. This kink happens on the electric field because of the slight delay when the charge is moved.

===Fields Made by Charges and Fields Made by Monopoles===
We can differentiate fields made by charges and the ones made by magnetic monopoles. (7)
For fields made by charges, when the charge is
*at rest, E=1/r^2 and B=0
*constant speed, E=1/r^2 and B=1/r^2
*accelerating, E=1/r and B=1/r

For fields made by magnetic monopoles, the first point would have E and B switched.

==The EM Spectrum==

EM spectrum is a span of enormous range of wavelengths and frequencies. The EM spectrum is generally divided into 7 different regions, in order of decreasing wavelength and increasing energy and frequency. It ranges from Gamma rays to Long Radio Waves. Following are the lists of waves:
* Gamma rays
* X-rays
* UV rays
* Visible Light
* Infrared Rays
* Microwave
* Radio, TV
* Long radio waves

[[File:em-spectrum.jpg]]

Although all these waves do different things, there is one thing in common : They all travel in waves.

[[File:spectrum_Properties.png]]

'''Infrared radiation''' can be released as heat or thermal energy. It can also be bounced back, which is called near infrared because of its similarities with visible light energy. Infrared Radiation is most commonly used in remote sensing as infrared sensors collect thermal energy, providing us with weather conditions.

[[File:pic_snap_girl.png]]

'''Visible Light''' is the only part of the electromagnetic spectrum that humans can see with a naked eye. This part of the spectrum includes a range of different colors that all represent a particular wavelength. Rainbows are formed in this way; light passes through matter in which it is absorbed or reflected based on its wavelength. As a result, some colors are reflected more than other, leading to the creation of a rainbow.

[[File:pyramid123.jpg]]

[[File:rainbow.jpg]]

==Waves and Fields==

As we learned in class, electric field is produced when an electron is accelerating. Likewise, EM radiation is created when an atomic particle, like an electron, is accelerated by an electric field. The movement like this produces oscillating electric and magnetic fields, which travel at right angles to each other in a bundle of light energy called a photon. Photons travel in a harmonic wave at the fastest speed possible in the universe.

[[File:waves image.jpg]]

Electromagnetic waves are formed when an electric field couples with a magnetic field. Magnetic and electric fields of an electromagnetic wave are perpendicular to each other and to the direction of the wave.

A wavelength (in m) is the distance between two consecutive peaks of a wave. Frequency is the number of waves that form in a given length of time. A wavelength and frequency are interrelated. A short wavelength indicates that the frequency will be higher because one cycle can pass in a shorter amount of time. Likewise, a longer wavelength has a lower frequency because each cycle takes longer to complete.

[[File:waves_1.jpg]]

Waves can be classified according to their nature:
* Mechanical waves
* Electromagnetic waves

'''Mechanical Waves'''

Mechanical waves require a medium (matter) to travel through.
Examples are sound waves, water waves, ripples in strings or springs.

''Water Waves''
[[File:waterwaves.gif]]

''Sound Waves''
[[File:loudspeaker-waveform.gif]]

'''Electromagnetic Waves'''

Electromagnetic waves do not require a medium (matter) to travel through - they can travel through space.
Examples are radio waves, visible light, x-rays.

''X-RAYS''
[[File:x-rays.jpeg]]

''Radio Waves''
[[File:facts-about-radio-waves.jpg]]

''Visible Lights''
[[File:visible-spectrum123.jpg]]

==A Mathematical Model==

The position of the particle is defined by a sine wave:

'''y = ymaxsin(wt)'''

Where w is the angular frequency

'''Amplitude'''

Amplitude is the distance from the maximum vertical displacement of the wave to the middle of the wave. The Amplitude of the sinusoidal Wave is the height of the peak in the wave measured from the zero line. This measures the magnitude of oscillation of a particular wave. The Amplitude is important because it tells you the intensity or brightness of a wave in comparison with other waves.

'''Period'''

The period of the wave is the time between crests in seconds(s).

T = 2pi/w-----(units of seconds)

'''Frequency'''

Frequency is the number of cycles per second, and is expressed as sec-1 or Hertz(Hz). Frequency is directly proportional to energy and can be express as "

'' E = hv ''
where E is energy, h is Planck's constant ( 6.62607*10^-34J) and v is frequency

f = 1/T
f = w/2pi----(Units Hertz)

'''Wavelength'''

Wavelength is the distance between crests in meters. Wavelength is equal to the speed of light times frequency. Longer wavelength waves such as radio waves carry low energy; this is why we can listen to the radio without any harmful consequences. Shorter wavelength waves such as x-rays carry higher energy that can be hazardous to our health.

[[File:shortlongwavelength.jpg]]

'''Wavelength and Frequency'''

The speed of light is the multiplication of the wavelength and frequency.

''c=λν ''

[[File:visible_EM_modes.png]]

This diagram shows all properties of waves:

[[File:wave_props.gif]]

'''ENERGY FLUX'''

Is defined by the following equation:

S = (1/u0)*(E x B) in W/m^2
where B = E/c
where c = speed of light

[[File:energy_flux.gif]]

==Connectedness: X-Rays==

Electromagnetic Radiation while commonly thought of as only including visible light, radio waves, UV waves, and gamma rays; also include X-rays. In 1895, X-rays were initially discovered by William Roentgen, who accidentally fell upon the most important discovery about his life (Figure 1). Roentgen was already working on cathode rays, and because of a fluorescent glow that occurred during his experiments, covered his experimental apparatus with heavy black paper. However, when he did this, he discovered a glow coming from a screen several feet away. Through many more experiments, he discovered that a new type of energy, not cathode rays, were the cause of the glow. He named them “x-rays” and received the 1901 Nobel Prize in Physics. Roentgen never patented his monumental discovery and as a result, numerous researchers set out to find a multitude of uses and capitalize on his work.

Primarily, people could now view objects that were hidden from plain view (i.e. scanners in airports). While X-rays are now used in 100’s of professions (security, chemistry, art galleries), its most important function is to view bones to determine abnormalities in humans. In fact, one of Roentgen’s first x-rays was of his wife’s hand (Figure 2). X-rays fall under the scope of electromagnetic radiation because, like all E.R. waves, it is comprised of photons. X-rays have wavelengths between 0.01 to 10 nanometers and fall between UV and Gamma Waves on the E.R. spectrum (Figure 3).
There are two main methods in which an x-ray may be formed. Both require a vacuum-filled tube called an x-ray tube (Figure 4). With an anode on one end and a cathode on the other, an electric current is applied and a high energy electron is projected from the cathode, through the vacuum, and at the anode. In the characteristic x-ray generation approach, the electron from the cathode collides with an inner shell electron on an atom on the anode (Figure 5). Both of these electrons are ejected from the atom and an outer shell electron takes the place of the inner shell one. Because the outer electron must have a lower energy to fill the inner shell hole, it releases a photon with the equivalent energy of the difference between the two energy levels in the atom. This photon is the x-ray that is used to view objects such as bones.

In the Bremsstrahlung x-ray generation method, the electron from the cathode is slowed as it passes the nucleus of an atom at the anode (Figure 6). As it slows and its path is changed, the loses energy (kinetic energy). This energy is also released as a photon which is subsequently called an x-ray.
Depending on the voltage and current of the tube and the material of the anode, different types (as in wavelengths and energy) of x-rays can be produced and each one. However, all X-rays will continue to pass through objects until it reaches a material dense that stops it. However, density of the material required depends on the energy of the x-ray. For example, during a medical x-ray, x-rays of a certain energy will pass through soft tissue (skin, organs, etc) but not through bones. The x-rays that pass through the soft tissue will strike the screen and the absence of the x-rays absorbed by the bones will cause a negative space on the screen. The areas where x-rays do not strike will form the image of the bone. While the principles remain the same, x-ray machines today use incredible sophisticated technology to specify the type of x-ray they want and have greatly increased in accuracy since Roentgen’s initial discovery.

Figure 1:
[[File:Monali_Figure_1.jpg]]

Figure 2:
[[File:Monali_Figure_2.jpg]]

Figure 3:
[[File:Monali_Figure_3.png]]

Figure 4:
[[File:Monali_Figure_4.png]]

Figure 5:
[[File:Monali_Figure_5.png]]

Figure 6:
[[File:Monali_Figure_6.png]]

*Information and photographs are pulled from references 1 through 5 cited below*

==History==

Already, during the Ancient Greek and Roman times, light was studied as the presence of deflection and refraction were noticed.
Electromagnetic radiation of wavelengths in the early 19th century. The discovery of infrared radiation is ascribed to astronomer William Herschel, who published his results in 1800 before the Royal Society of London. Herschel used a glass Triangular prism (optics)|prism to refract light from the Sun and detected invisible rays that caused heating beyond the red part of the spectrum, through an increase in the temperature recorded with a thermometer. These "calorific rays" were later termed infrared.

In 1801, Rohann Ritter, discovered the presence of ultraviolet light using salts. It was known that light could darken some silver halides and while doing so, he realized that the region beyond the violet bar (therefore ultraviolet) was more effective in changing the color of the halides.
However,in 1864, while summarizing the theories of his time accumulating into his famous set of Maxwell equations, James Clerk Maxwell managed to deduce the speed of light being around 3e8 meters per second. This was instrumental in creating the rest of the spectrum.

In 1887-1888 Physicist Heinrich Hertz not only tried to measure the velocity and frequency of electromagnetic radiation waves at other parts of the known spectrum of the time, but he was also able to prove that Maxwell's findings were correct. He did this on the microwave radiation as well.

The discovery of X-rays occurred in 1895 by Wilhelm Rontgen when his barium platinocyanide detector screen began to glow under the presence of a discharge that passed through a cathode ray tube although the latter was completely covered. Once he determined its possible use, he tried to look at his wife's hand using this new discovery. However x-ray spectroscopy was not institutionalized until later by Karl Manne Siegbahn.

In 1900, Paul Villard discovered Gamma rays although he initially thought that they were particles similar to alpha and beta particles which were emitted during radiation. These 'particles' were later proven to be part of the electromagnetic spectrum.

===Practice Problems (new section by Zoila)===

Before we get to practicing some problems from this topic here are some helpful slides from class with equations and explanations.[[File:Thingy.jpg]]

Here are some examples using these equations:
[[File:Qo.jpg]]

1.)

==References==
1. Elert, Glenn. "X-rays." X-rays – The Physics Hypertextbook. N.p., n.d. Web. 08 Apr. 2017.
http://physics.info/x-ray/

2."X-rays." X-rays. N.p., n.d. Web. 08 Apr. 2017. http://www.physics.isu.edu/radinf/xray.htm

3. "Basics of X-ray PhysicsX-ray production." Welcome to Radiology Masterclass. N.p., n.d. Web. 08 Apr. 2017. http://www.radiologymasterclass.co.uk/tutorials/physics/x-ray_physics_production#top_2nd_img

4. "X-Rays." Image: Electromagnetic Spectrum. N.p., n.d. Web. 08 Apr. 2017. https://www.boundless.com/physics/textbooks/boundless-physics-textbook/electromagnetic-waves-23/the-electromagnetic-spectrum-165/x-rays-597-11175/images/electromagnetic-spectrum/

5. "This Month in Physics History." American Physical Society. N.p., n.d. Web. 08 Apr. 2017. https://www.aps.org/publications/apsnews/200111/history.cfm

6. Editors, Spectroscopy. “The Electromagnetic Spectrum: A History.” Spectroscopy Home, 27 Oct. 2017, www.spectroscopyonline.com/electromagnetic-spectrum-history?id=&sk=&date=&&pageID=4.

7. Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interaction II: Electric & Magnetic Interactions, Version 1.2. John Wiley & Sons, 2003.

Faraday's Law

2023-11-12T01:53:12Z

Daniel72: /* Curly Electric Field */ Modified image since the old version was very vague as to why the induced field was flipping. This one is clear about the direction and increasing/decreasing.

Claimed by Ananya Ghose Fall 2021

Note to editors: need a computational model

Faraday's Law
focuses on how a time-varying magnetic field produces a "curly" non-Coulomb electric field, thereby inducing an emf.

==Faraday's Law==

Faraday's Law summarizes the ways voltage can be generated as a result of a time-varying magnetic flux. And it gives a way to connect the magnetic and electric fields in a quantifiable way (will elaborate later). Faraday's law is one of four laws in Maxwell's equations. It tells us that in the presence of a time-varying magnetic field or current (which induces a time-varying magnetic field), there is an emf with a magnitude equal to the change in magnetic flux. It serves as a succinct summary of the ways a voltage (or emf) may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with the magnetic field.

==Curly Electric Field==

[[File:Newcurly.png]]

===Mathematical Model===

'''Faraday's Law'''

emf = <math>{\frac{-d{{Phi}}_{mag}}{dt}}</math>

where emf = <math>\oint\vec{E}_{NC}\bullet d\vec{l}</math> and <math>{{Phi}}_{mag}\equiv\int\vec{B}\bullet\hat{n}dA</math>

In other words: The emf along a round-trip is equal to the rate of change of the magnetic flux on the area encircled by the path.

Direction: With the thumb of your right hand pointing in the direction of the ''-dB/dt'', your fingers curl around in the direction of Enc.

The meaning of the minus sign: If the thumb of your right hand points in the direction of ''-dB/dt'' (that is, the opposite of the direction in which the magnetic field is increasing), your fingers curl around in the direction along which the path integral of electric field is positive. Similarly, the direction of the induced current can be explained using Lenz's Law. Lenz's law states that the induced current from the non-Coulombic electric field is induced in such a way that it produces a magnetic field that opposes the first magnetic field to keep the magnetic flux constant.

'''Formal Version of Faraday's Law'''

<math>\oint\vec{E}_{NC}\bullet d\vec{l} = {\frac{-d}{dt}}\int\vec{B}\bullet\hat{n}dA</math> (sign given by right-hand rule)

===Fiding the direction of the induced conventional current===
To find the direction of the induced conventional current by the change in the magnetic flux one must find the direction of the Non-Coulomb electric filed generated by the change in flux as the conventional current is the direction of the Non-Coulomb electric field.
To find the direction of the the Non-Coulomb Electic field, one must find the direction of <math> \frac{-dB}{dt} </math>. Do this using the change in magnetic field as the basis of finding the <math> \frac{-dB}{dt} </math>.

As stated previously the negative sign in front of the change in magnetic flux in the Law is a representative of Lenz's law or in other words, it's there to remind us to apply Lenz's law. Lenz's law is basically there to make us abide by the law of conservation of energy. That said, thinking in terms of conservation of energy provides the simplest way to figure out the direction of the Non-Coulomb electric field.
The external magnetic field induces the Non-Coulomb electric field which drives the current which in turn creates a new magnetic field which we will call the induced magnetic field. This is the magnetic field whose direction we can deduce which in turn will help us find the direction of the current.
The easiest way to do this is to imagine a loop of wire with and an external magnetic field perpendicular to the surface of the plane of the loop. There is a change in magnetic flux generated by the change in the magnitude of the magnetic field. vector for the initial external magnetic field and a vector for the final magnetic field. Then, draw the change in magnetic field vector, <math> \Delta \mathbf{B} </math>, and then the negative vector of that change in magnetic field gives <math> \frac{-dB}{dt} </math>:

[[File:neg_change_B_dt.jpg]]

Pointing the thumb of your right hand in the direction of <math> \frac{-dB}{dt} </math> allows you to curl your fingers in the direction of <math> \mathbf{E_{NC}} </math>.

In this chapter we have seen that a changing magnetic flux induces an emf:

[[File:tips5.png]]

according to Faraday’s law of induction. For a conductor which forms a closed loop, the
emf sets up an induced current ''I =|ε|/R'' , where ''R'' is the resistance of the loop. To
compute the induced current and its direction, we follow the procedure below:

1. For the closed loop of area on a plane, define an area vector A and let it point in
the direction of your thumb, for the convenience of applying the right-hand rule later.
Compute the magnetic flux through the loop using

[[File:tips4.png]]

Determine the sign of the magnetic flux [[File:tips3.png]]

2. Evaluate the rate of change of magnetic flux [[File:tips2.png]] . Keep in mind that the change
could be caused by

[[File:tips.png]]

Determine the sign of [[File:tips2.png]]

3. The sign of the induced emf is the opposite of that of [[File:tips2.png]]. The direction of the
induced current can be found by using Lenz’s law or right-hand rule (discussed previously).

==Computational Model==
The following simulations demonstrate Faraday's Law in action.

==More on Faraday's Law==

Moving a magnet near a coil is not the only way to induce an emf in the coil. Another way to induce emf in a coil is to bring another coil with a steady current near the first coil, thereby changing the magnetic field (and flux) surrounding the first coil, inducing an emf and a current. Also, rotating a bar magnet (or coil) near a coil produces a time-varying magnetic field in the coil since rotating the magnet changes the magnetic field in the coil. The key to inducing the emf in the second coil is to change the magnetic field around it somehow, either by bringing an object that has its own magnetic field around that coil, or changing the current in that object, changing its magnetic field.

Faraday's law can be used to calculate motional emf as well. A bar on two current-carrying rails connected by a resistor moves along the rails, using a magnetic force to induce a current in the wire. There is a magnetic field going into the page. One way to calculate the motional emf is to use the [http://www.physicsbook.gatech.edu/Motional_Emf magnetic force], but an easier way is to use Faraday's law.

Faraday's law, using the change in magnetic flux, can be used to find the motional emf, where the changing factor in the magnetic flux is the area of the circuit as the bar moves, while the magnetic field is kept constant.

[[File:motionalemf.jpg]]

==Examples==

===Simple===

[[File:solenoid.ring.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P12 (4th ed)''.

The solenoid radius is 4 cm and the ring radius is 20 cm. B = 0.8 T inside the solenoid and approximately 0 outside the solenoid. What is the magnetic flux through the outer ring?

''Solution:''

Because the magnetic field outside the solenoid is 0, there is no flux between the ring and solenoid. So the flux in the ring is due to the area of the solenoid, so we use the area of the solenoid to find the flux through the outer ring rather than the area of the ring itself:

<math> \phi = BAcos(\theta)</math>

<math>= (0.8 T)(\pi)(0.04 m)^2cos(0) </math>

<math>= 4.02 x 10^{-3} T*m^2 </math>

===Middle===

[[File:rectanglecoilsolenoid.jpg|center|alt=Diagram for simple example]]

''Adapted from the'' Matter & Interactions ''textbook, variation of P27 (4th ed)''.

A very long, tightly wound solenoid has a circular cross-section of radius 2 cm (only a portion of the very long solenoid is shown). The magnetic field outside the solenoid is negligible. Throughout the inside of the solenoid the magnetic field ''B'' is uniform, to the left as shown, but varying with time ''t: B'' = (.06+.02<math>t^2</math>)T. Surrounding the circular solenoid is a loop of 7 turns of wire in the shape of a rectangle 6 cm by 12 cm. The total resistance of the 7-turn loop is 0.2 ohms.

(a) At ''t'' = 2 s, what is the direction of the current in the 7-turn loop? Explain briefly.

(b) At ''t'' = 2 s, what is the magnitude of the current in the 7-turn loop? Explain briefly.

''Solution''

'''(a)''' The direction of the current in the loop is clockwise.

'''(b)'''

B(t) = (.06+.02<math>t^2</math>)

A = (π)(0.02 m)^2 = .00126 <math>m^2</math>

<math>|{ε}| = AN\frac{dB(t)}{dt}</math>

<math>|{ε}|</math> = (.00126 <math>m^2</math>)(7)<math>\frac{d(.06+.02t^2)}{dt}</math> = (.00882)(.02)(2t) = .0003528t

'''At ''t'' = 2 s:'''

<math>|{ε}|</math> = .0003528(2) = .0007056 V

<math>i = \frac{{ε}}{R}</math>

<math>i = \frac{{.0007056 V}}{0.2 ohms}</math> = '''.00353 A'''

===Difficult===

[[File:difficultfaraday.png]]

A square loop (dimensions L⇥L, total resistance R) is located halfway inside a region with uniform magnetic field B0. The magnitude of the magnetic field suddenly begins to increase linearly in time, eventually quadrupling in a time T.

'''(a) What current (magnitude and direction), if any, is induced in the loop at time T?
'''

<math> |emf| = \frac{-{Φ}_{B}}{Δt} = \frac{A(B_f - B_i)}{T} = \frac{L^2(4B_o - B_o)}{T} = \frac{3B_oL^2}{T}</math>

emf = IR = <math>\frac{3B_oL^2}{TR}</math>

'''(b) What net force (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> F_{top} </math> and <math> F_{bottom} </math> cancel out.
<math> F_{left} </math> = 0 because the left side is out of <math> \vec{B} </math> region.

<math> \vec{F}</math> = <math> \vec{F}_{right} </math> = I <math> \vec{L} \times \vec{B} = (ILB)[(\hat{y} \times - \hat{z} )] = \frac{3B_oL^2}{TR}(4B_o L)(- \hat{x}) = \frac{3{B_o}^2 L^3}{TR}(- \hat{x})</math>

'''(c) What net torque (magnitude and direction), if any, is induced on the loop at time T?
'''

<math> \vec{τ} = \vec{μ} \times \vec{B} = 0 </math> because <math>\vec{μ}</math> and <math>\vec{B}</math> are anti-parallel.

==Connectedness==

Faraday's Law is one of Maxwell's equations which describe the essence of electric and magnetic fields. Maxwell's equations effectively summarize and connect all that we have learned throughout the course of Physics 2.

As an electrical engineer, Faraday's Law is relevant to my major.

== Faraday’s Law Applications ==

Physics 2 content has a lot of important concepts that we as engineers can use to make our jobs easier. Whether it be a direct application of a rule or some derivation of a rule. I know I personally struggle with a concept until I get a concrete real life application that I can see the material applied in. This section of the page will discuss how Faraday’s law is applied to concepts that you as students maybe more familiar with your day to day life.

== Hydroelectric Generators ==
Generators create energy by transforming mechanical motion into electrical energy, but hydroelectric generators use the power of falling water to turn a large turbine which is connected to a large magnet. Around this magnet is a large coil of tightly wound wire. The conceptual creation of electricity is the same as Faraday’s Law except alternating current is being produced, but the idea that a changing magnetic field in a coil of wire induces an electromotive force is still the same. The difference is the magnetic field changes sign and flips resulting in the same thing to occur in the induced EMF. Although the calculations here are slightly more difficult the concepts are the same.

== Transformers ==

Transformers use a similar concept for Faraday’s Law but it’s slightly different. The job of a transformer is to either step up or step down the voltage on the power line. Transformers have a constant magnetic field associated with it due to an iron core. The power supply voltage is adjusted by altering the number of turns of wire around the iron core which in turn alters the EMF of the electricity.

Cartoon of Hydroelectric Plant
https://etrical.files.wordpress.com/2009/12/hydrohow.jpg
Turbine Picture
http://theprepperpodcast.com/wp-content/uploads/2016/02/108-All-About-Hydro-Power-Generators-1054x500.jpg
Transformer Diagram https://en.wikipedia.org/wiki/Transformer#/media/File:Transformer3d_col3.svg

==History==

In 1831, eletromagnetic induction was discovered by Michael Faraday.

===Faraday's Law Experiment ===

[[File:experiment.png]]

Faraday showed that no current is registered in the galvanometer when bar magnet is
stationary with respect to the loop. However, a current is induced in the loop when a
relative motion exists between the bar magnet and the loop. In particular, the
galvanometer deflects in one direction as the magnet approaches the loop, and the
opposite direction as it moves away.

Faraday’s experiment demonstrates that an electric current is induced in the loop by
changing the magnetic field. The coil behaves as if it were connected to an emf source.
Experimentally it is found that the induced emf depends on the rate of change of
magnetic flux through the coil.

Test it out yourself [https://phet.colorado.edu/en/simulation/faradays-law here]

==See also==
===Further Readings===

''Matter and Interactions, Volume II: Electric and Magnetic Interactions, 4th Edition''

''The Electric Life of Michael Faraday'' (2009) by Alan Hirshfield

''Electromagnetic Induction Phenomena'' (2012) by D. Schieber

===External Links===

https://www.youtube.com/watch?v=KGTZPTnZBFE

https://www.nde-ed.org/EducationResources/HighSchool/Electricity/electroinduction.htm

http://www.famousscientists.org/michael-faraday/

http://www.bbc.co.uk/history/historic_figures/faraday_michael.shtml

==References==

''Matter and Interactions, 4th Edition''

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

https://files.t-square.gatech.edu/access/content/group/gtc-970b-7c13-52a7-9627-cdc3154438c6/Test%20Preparation/Old%20Test/2212_Test4_Key-1.pdf

https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

File:Newcurly.png

2023-11-12T01:51:31Z

Daniel72: Daniel72 uploaded a new version of File:Newcurly.png

== Summary ==
New version of the old curly.png which better explains how the orientation of the field (north side of magnet closer vs south side) and its change over time (increasing vs decreasing) change the direction of the induced current. Old version was very vague and could lead to confusion.

File:Newcurly.png

2023-11-12T01:47:30Z

Daniel72: New version of the old curly.png which better explains how the orientation of the field (north side of magnet closer vs south side) and its change over time (increasing vs decreasing) change the direction of the induced current. Old version was very vague and could lead to confusion.

Magnetic Torque

2023-11-12T01:12:59Z

Daniel72: /* Magnetic Dipole Moment */ Clarified the definition of the direction of Magnetic Dipole Moment. It does not in fact point in the direction of the magnetic field (at least not always), but along the magnetic dipole.

Magnetic torque is induced when a magnetic field causes a current carrying coil of wire to twist.
[[File:torqueexample.png|thumb|Example of Magnetic Torque]]
==The Main Idea==

The idea behind this concept is that as current flows through a wire, a magnetic field is produced. This magnetic field causes a force to act upon the wire causing it to twist. An example of this phenomenon is the movement of a compass needle by the Earth's magnetic field. Another example is a hanging coil that twists in the direction of the magnetic field of a bar magnet.

The magnetic torque acts on the dipole, and it is highly dependent on the magnetic moment and external magnetic field.

Several factors besides the magnetic moment and external magnetic field can affect the magnetic torque. In a loop or other three dimensional object the orientation of the object relative to the magnetic field highly affects the torque.

Through the following general example you can see how this phenomena occurs:
[[File:Magnetic_Torque_Main_Idea_1.png]]

On the sides h, the magnetic force is horizontal pointing outwards causing the loop to stretch; while on the sides of length w the magnetic forces are horizontal and tend to make the loop twist on the axle. This causes the loop to rotate counterclockwise. When the plate of the loop is perpendicular to the magnetic field don't exert any twist.

There are two configurations: Stable and Unstable

[[File:Magnetic_Torque_Main_Idea_2.png]]

In the stable configuration, magnetic forces will twist the loop back up to the horizontal plane. In the unstable configuration, small displacement away from the horizontal leads to magnetic forces that rotate it even farther out of the plane.

This relationship can be seen in this video:
[https://www.youtube.com/watch?v=E-3yQqgu8OA]

Here is a video on Asymmetric Magnet Torque
[http://www.youtube.com/watch?v=LD6TX5IH5po Asymmetric Magnet Torque]
==A Mathematical Model==
The overarching equation that encapsulates this physical phenomena is as follows:
: <math> \boldsymbol{\tau} = \boldsymbol{\mu} \times\mathbf{B}</math>

where:

'''τ''' is the variable describing torque

'''μ''' is the magnetic dipole and can be found using many expressions including that of a wire which relates magnetic dipole to the current in the wire multiplied by its cross sectional area. For a magnet, this quanity is not easily derived, and is a little outside the scope of this discussion. This quanitity is usually given in the problem statement. However, for a video that helps describe the magnetic dipole moment of a magnet: [https://www.youtube.com/watch?v=lOSmfcS1Vrg]

'''B''' is the magnetic field

The torque provided by each of the magnetic forces around the axle is equal to the distance from the axle times the component of the force perpendicular to the lever. Twist applied is due to the w - sides of the loop where torque acts out of the page. This causes a clockwise twist.

<math> F_{perpendicular} = I*w*B*sin(x) </math> where the arm is equal to h/2, each side exerts a force <math> F = 2(I*w*B*sin(x))(h/2) </math>

<math> τ = I*w*B*sin(x) </math> and <math> µ = I*w*h </math>

<math> τ = µ x B = µ*B*sin(x) </math>

The right hand rule for the direction of torque is as follows: the fingers of your right hand curl in the direction the loop will rotate, and your thumb will point the the direction of torque. The direction of the torque vector will be along the axle around which the loop rotates. For a more in depth explanation of the right hand rule see [[Right-Hand Rule]]

===Magnetic Dipole Moment===

The magnetic dipole moment of a current carrying loop of wire, '''µ''', is defined as a vector pointing in the direction of the magnetic dipole (South to North) perpendicular to the plane created by the wire loop, and its direction is determined by using the right hand rule (fingers curling along the direction of the current, thumb pointing in dipole direction.)

<math> µ = I*A = I*w*h </math>

The coil tends to twist in a direction to make '''µ''' line up with '''B'''.

[[File:Magnetic_Torque_Mathematical_Model.png]]

===Units Discussion===
'''τ''' has units of N*m

'''µ''' has units of A*m^2

'''B''' has units of tesla or T

From this, it must be that one N*m (which interestingly defines work) is equal to one tesla * A*m^2. From a discussion of units alone, it is important to think about what sorts of questions the professor might ask, meaning questions could include an analyses of the work that must be added to a system to keep it stationary for example.

==A Computational Model==

Click here to view the PHET Interactive Model created by the University of Colorado

[https://phet.colorado.edu/en/simulation/legacy/magnet-and-compass PHET Interactive Magnet and Compass Model]

==Examples==
Essentially, there are only a few categories of questions that can be asked relating to magnetic torque. These questions include a simple computation of magnetic torque given the dipole moment of a magnet, and the magnetic field being applied to the observation location. In this situation, you can either utilize a simple cross product, as in the equation listed above, or if the values are given as scalars, and it is known that they are perpendicular to each other in direction, you can utilize the equation: <math> |τ| = |µ| * |B|cos(90) = |µ| * |B| </math>. This is the essential question involving the equation listed above for magnetic torque. However, the professor can also ask questions relating to material learned from physics 1 involving angular frequencies and other products of angular momentum.

[https://www.youtube.com/watch?v=xER1_SYql44 Torque on Current Carrying Loop]

===Simple===

A bar magnet whose magnetic dipole moment is <3, 0, 1.8> A · m2 is suspended from a thread in a region where external coils apply a magnetic field of <0.6, 0, 0> T. What is the vector torque that acts on the bar magnet?

[[File:SimpleWikiProb.JPG]]

===Middling===
A bar magnet whose magnetic dipole moment is 14 A · m2 is aligned with an applied magnetic field of 5.4 T. How much work must you do to rotate the bar magnet 180° to point in the direction opposite to the magnetic field?

[[File:MiddleWikiProb.JPG]]

===Difficult===

A cylindrical bar magnet whose mass is 0.09 kg, diameter is 1 cm, length is 3 cm, and whose magnetic dipole moment is <4, 0, 0> A · m2
is suspended on a low-friction pivot in a region where external coils apply a magnetic field of <2.0, 0, 0> T. You rotate the bar magnet slightly in the horizontal plane and release it. (For small angles in radians, assume sin(θ) ≈ θ.)

(a) What is the angular frequency of the oscillating magnet?

(b) What would be the angular frequency if the applied magnetic field were <4.0, 0, 0> T?

[[File:DifficultWikiProb.JPG]]

A detailed description and symbolic representation of magnetic torque can be seen here:
[https://www.youtube.com/watch?v=K1FEepXKETM Magnetic Torque and Magnetic Dipole Moment]

==Connectedness==

[[File:Compass.jpg|thumb|A standard compass http://helenotway.edublogs.org/2011/01/02/different-compass-point-same-ultimate-direction/]]

Utilizing a compass is a basic survival need and it just so happens to depend on the torque produced by the Earth's magnetic field. As a Biology major, field work is a large part of what I do, especially studying ecological systems and different habitats. In order to navigate in unfamiliar locations, such as deserts and dense tropical forests, scientists rely heavily on basic survival skills and this includes the use of compasses and maps. Physics, biology, and chemistry make up part of the science family and each heavily depends on the other, this is why it is important to study each one to bridge the relationship.

First paragraph of "Connectedness" written by Demetria Hubbard 2015

The Earth has a complex magnetic field and magnetic dipole moment that creates a magnetic torque. The necessity of all three of these magnetic properties is rarely known; however, all three are essential for life on earth. Earth's magnetic field serves to deflect most of the solar wind, so without the magnetic properties of the earth, the charged solar wind would have stripped the ozone layer from earth which would have exposed everything on earth to dangerous UV radiation.

[[File:Earth's magnetic field, schematic.svg|thumb|right|Earth's magnetic field, schematic]]

One interesting development in the field of magnetic torque is the experimentation, and initial prototyping of magnetic gears for application in a wide variety of industries, but that has a main focus in the wind turbine industry. The issue with strictly mechanical gearing today is in a high stress situation, the “teeth” or connection between gears, will fracture as a result of being over torqued. This results in a very powerful stall out that can gravely damage the broader mechanics of the instrument that the gears are in. Magnetic gears provide an interesting solution to the problem because there is no “physical” interaction between gear faces, only magnetic forces. This mitigates the stalling issue and provides a higher torque range by which machines utilizing this technology can operate. Just to give a specific example of this application, in the oil drilling industry, specifically where mud motors are applied to prospect oil, there is an incredible amount of power that must be applied via torque translation from the power section to the drill bit. An issue often seen is the wearing down of gears along the drill chain as a result of lubrication leaking, and rubbing of two components together, leading to stall outs which can damage the drill overall. To counteract this problem, research has been started to develop magnetic transmission sections to transmit the torque provided by the power section to the drill bit with minimal part damage due to minimal rubbing of components. The introduction of the magnetic gear will also mitigate the cost of lubricants, which is a very high cost especially when expensive lubricants are required.

==History==

Refer to [[Magnetic Force]]

The great importance of magnetic torque that is used in compasses cannot be ignored. The history of the compass and earth's magnetic field are very valuable.
The tendency of a magnet to align itself was discovered by the Chinese about 2000 years ago. The magnetic compass became a valuable commodity to European navigators in the 12th century, and in 1600, William Gilbert published De Magnete, which concluded that the earth behaves as a giant magnet.
Several theories since then have been made to explain how a magnetic field is produced by the earth. The most accepted theory is that the energy from the radioactivity of the earth's core travels outwards as heat. This heat produces a thermal convection core that creates the earth's magnetic field.

== See also ==

* [[Torque]]
* [[Magnetic Field]]
* [[Magnetic Force]]
* [[Bar Magnet]]

===Further reading===

* Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 3rd ed. Hoboken, NJ: Wiley, 2011. Print.
* Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, p. 269, 1985.
* Griffiths, D. J. Introduction to Electrodynamics, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, p. 220, 1989.

===External links===

[http://scienceworld.wolfram.com/physics/MagneticTorque.html Magnetic Torque]

==References==

* [http://commons.wikimedia.org/wiki/File:Momento_torcente_magnetico.svg Torque Example]
* Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 3rd ed. Hoboken, NJ: Wiley, 2011. Print.
* "Magnet and Compass PHET Interaction Model." PhET. Ed. Chris Malley. University of Colorado, 2015. Web. 5 Dec. 2015. <https://phet.colorado.edu/en/simulation/legacy/magnet-and-compass>.
* Torque on Current-Carrying Loop in Magnetic Field. Doc Schuster. 23 Jan. 2013. Video. https://www.youtube.com/watch?v=xER1_SYql44
* http://helenotway.edublogs.org/2011/01/02/different-compass-point-same-ultimate-direction/
* Weisstein, Eric. "Magnetic Torque." Eric Weisstein's World of Physics. Wolfram Research, 1996. Web. 5 Dec. 2015. <http://scienceworld.wolfram.com/physics/MagneticTorque.html>.
* "Magnetic Torques and Amp's Law." Rochester Institute of Technology. Web. 5 Dec. 2015. <http://spiff.rit.edu/classes/phys213/lectures/amp/amp_long.html>.
* "Homework 11." WebAssign. Web. 5 Dec. 2015. <http://webassign.net/>.
* Magnetic Torque. Animations for Physics and Astronomy. 15 Feb. 2008. Video. https://www.youtube.com/watch?v=xER1_SYql44
* Digital image. N.p., n.d. Web. 17 Apr. 2016.
* "Discovery of the Earth’s Magnetic Field." GNS Science. N.p., n.d. Web. 17 Apr. 2016. <http://www.gns.cri.nz/Home/Our-Science/Earth-Science/Earth-s-Magnetic-Field/Discovery-of-the-Earth-s-magnetic-field>.
* "Magnetic Dipole Moment." Hyperphysics, n.d. Web. 17 Apr. 2016. <http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html>.
* Magnetic Torque and Magnetic Dipole Moment. AK Lectures. 7 Dec. 2013. Video. https://www.youtube.com/watch?v=K1FEepXKETM
* "Magnetism." DISCovering Science. Gale Research, 1996. Reproduced in Discovering Collection. Farmington Hills, Mich.: Gale Group. December, 2000. http://galenet.galegroup.com/servlet/DC/
* Jun 19, 2014 Leland Teschler | Machine Design. "Could Magnetic Gears Make Wind Turbines Say Goodbye to Mechanical Gearboxes?" Machine Design. Penton, 19 June 2014. Web. 27 Nov. 2016.

[[Category:Fields]]

Kirchoff's Laws

2023-11-11T16:35:50Z

Daniel72: /* The Main Idea */ While the equation for Loop 2 was mathematically consistent, flipping the signs (which is equivalent mathematically) aligns more with the explanation given by the paragraph below the three loop equations.

Edits: Regina Ivanna Gomez Quiroz, Spring 2023

[[File:Junction.gif|thumb|The node (or junction) rule states that the current flowing in is equal to the current that flows out.
I1 + I4 = I2 + I3 + I5]]

Kirchoff's Laws are two fundamental principles of electric circuits and are used to determine the behaviors of electric circuits and their components. They serve as a guide for how circuits will behave and are accurate for all DC and low-frequency AC circuits. These principles are used to measure voltage and current in [http://www.physicsbook.gatech.edu/Ammeters,Voltmeters,Ohmmeters voltmeters and ammeters].

Kirchoff's First Law, also known as Kirchoff's Node Rule or Kirchoff's Junction Rule, further exercises the law of Conservation of Charge and states that if current is constant, all the current that flows through one junction must be equal to all the current that flows out of the junction. This rule can be applied both to [[Electron and Conventional Current|conventional]] and [[Electron and Conventional Current|electron currents]]. This rule is not a fundamental principle, but rather a consequence of the fundamental principle of conservation of charge and the definition of steady-state. This law essentially states that the net amount of current entering a particular juncture in the system is equal to the net amount of current leaving the system. A mathematical way to think about is that the sum of the currents entering a point will be equal to the sum leaving it.

[[File:Visual_Model2.png|thumb|The Loop rule states that the sum of voltage around a loop is equal to 0.
LOOP 1: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CF} + \Delta {V}_{FA} = 0 </math>

LOOP 2: <math> \Delta {V}_{FC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} = 0 </math>

LOOP 3: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} + \Delta {V}_{FA} = 0 </math>]]

Kirchoff's Second Law, also known as Kirchhoff's Loop Rule or Kirchhoff's Voltage Law states that the sum of potential differences around a closed circuit is equal to zero. More simply, in a completed circuit, the voltages around a loop will sum to 0. This is because voltage is just energy per unit charge, and both energy and charge are conserved by fundamental laws. A good way to think about this is by attaching a positive value to the points in the loop that contribute to the net voltage, and by attaching a negative value to the resistors. The amount of voltage that enters the loop will be equal to the amount that is dissipated across the loop, thus ensuring that the net voltage is zero.

Note that this is only true when the magnetic field is neither fluctuating nor time-varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.

Both laws have limitations; most specifically, the node rule relies on the net charge in the system being constant, while loop rule depends on time-varying magnetic fields being confined to singular parts.

==The Main Idea==

'''Kirchoff's Node Rule'''

The node rule states that at any junction in an electrical circuit, the amount of current flowing into the junction is equal to the amount of current flowing out of the junction in steady-state.

In the steady state, for many electrons flowing into and out of a node,
* electron current: <math> net\ i_{in} = net\ i_{out},</math> where <math>i = nA\mu</math><math>E</math>
* conventional current: <math> net\ I_{in} = net\ I_{out},</math> where <math>I = |q|nA\mu</math><math>E</math>

'''Conservation of Charge'''

This rule is an application of the conservation of electric charge, basically that charge within a circuit cannot be created or lost. During the flow process around the circuit, there is no loss of any charge, thus the total current in any cross-section of the circuit is the same. If there are no nodes in the loop, the conventional current is the same throughout the loop. This also why the node rule relies on a constant charge throughout the system; if the charge was constantly changing, then the node rule will not apply. The node rule essentially states that the net current entering a juncture in the system will always be constant.

'''Kirchoff's Loop Rule'''

The loop rule simply states that in any round trip path in a circuit,
[[Electric Potential]] equals zero. This applies through any round trip path; in more complex circuits, there can be
multiple round trip paths. This principle is an application of the conservation of energy, specifically within a circuit.
This principle is often used to solve for resistance or current passing through of light bulbs and other resistors, as well as the capacitance or charge of capacitors in a circuit.

[[File:loopexample.png]]

LOOP 1: <math> \Delta {V}_1 = emf - I_1R_1 = 0 </math>

LOOP 2: <math> \Delta {V}_2 = I_1R_1 - I_2R_2 = 0 </math>

LOOP 3: <math> \Delta {V}_3 = emf - I_2R_2 = 0 </math>

To figure out the sign of the voltages, act as an observer walking along the path. Start at the negative end of the emf and continue walking along the path. The emf will be positive in the loop rule because you are moving from low to high voltage. Once you reach a resistor or capacitor, this will be negative in the loop rule equation because it is high to low voltage. Continue along the path until you return to the starting position.

'''Bringing Both Laws Together'''

Find all possible node and loop equations for the following circuit.

[[File:Non_steady_state_image_TWO.png|400 px]]

You should have 3 Loop Equations and 2 Node equations.

Some examples:

<math> I_1= I_2+I_3 </math>

<math> I_1*R_1 + emf_1 + emf_3 + I_3*R_4 = 0 </math>

'''Matrices in Conjunction with Kirchhoff's Laws'''

After creating loop equations for all possible loops in a circuit using the node rule and loop rule, you can input the coefficients of the equations into a matrix. After inputting it into the matrix, you can use linear algebra and row reduction in order to solve for the current or resistances of the different parts of the circuit. This format and method of understanding complex circuits are especially useful for circuits with a large number of loops as it makes the algebra involved in getting the currents much simpler and straightforward. Simply creating the equations for every loop possible and every node and then inputting the coefficients into a row reduction calculator can give you all the values for I and/or R that you might need.

After finding the loop equations you can use a matrix calculator as well to further simplify your task!: [http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=rref Online Matrix Row Reduction Calculator]

An example of how you would format the matrix of a Kirchhoff's law problem is shown below:

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& & & | EMF \\
& & & | EMF
\end{bmatrix}.</math>

Simply place to coefficients or resistances of each of the different resistors as separate elements in the columns of the matrix while keeping track of which resistance pertains to which part of the loop and place the EMF of that loop as the solution column in the augmented matrix. Then after row reducing, each of the resistors should be simplified so that there is a 1 in at least 1 row for every resistor and the current at the end of that row is the current for the resistor that is in the same column as the 1 in the row of the current.

For example in the matrix below the current going through R2 is 1/15 amps:

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& 1 & & | 1/15 \\
& & & | EMF
\end{bmatrix}.</math>

'''Limitations'''

The Node rule assumes that current flows only in conductors and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for high-frequency AC circuits. In other words, the node is valid only if the total electric charge, <math>Q</math>, remains constant in the region being considered. In practical cases, this is always so when the node rule is applied at a point.

The Loop rule is based on the assumption that there is no fluctuating magnetic field linking the closed-loop.
This is not a safe assumption for high-frequency AC circuits. In the presence of a changing magnetic field, the electric field is not a conservative vector field. Therefore, the electric field cannot be the gradient of any potential. That is to say, the line integral of the electric field around the loop is not zero, directly contradicting KVL.

Understanding these limitations is essential to understanding how motional emf, motors, and generators work.

===A Mathematical Model===

'''Kirchoff's Node Rule'''

[[File:kircho2.gif|right]]
The node rule can be stated as:
:<math> \sum \Delta{I} = 0 </math> 
where <math>I</math> stands for the current of the individual parts or wires in a circuit and the sign of the current that flows into the junction is opposite of the current that flows out of the junction. This simply supports the idea that charge cannot be created or destroyed because the total current must remain equal regardless of the path it takes.
:<math> \sum_{k=1}^n I_k = 0 </math> 
where <math>n</math> is the total number of branches with current flowing through the node, as well as along any node in a circuit:

:<math> \Delta {I}_{1} + \Delta {I}_{2} + \space.... = 0 </math>

or more generally:

:<math> \sum_{k=1}^n \tilde{I_k}</math> = 0 

Another definition is simply a summation rule:
"<math> \sum I(in) = \sum I(out)</math>
'''Kirchoff's Loop Rule'''

A mathematical representation is:

:<math> \sum \Delta{V} = 0 </math> 
where <math>V</math> stands for the voltage of the individual parts or wires in a circuit and the sign of the voltage that flows clock-wise is opposite of the current that flows counterclockwise. This simply supports the idea that energy cannot be created or destroyed because the total current must remain equal regardless of the path it takes.

:<math> \sum_{i=1}^n {V}_{i} = 0 </math>

where <math>n</math> is the number of voltages being measured in the loop, as well as

:<math> \Delta {V}_{1} + \Delta {V}_{2} + \space.... = 0 </math>

along any closed path in a circuit. The voltages may also be complex:

:<math>\sum_{k=1}^n \tilde{V}_k = 0</math>

'''Matrix Intrepretation'''

Using matrices can simplify the process of using node and loop rules in order to find out different parts of a complex circuit. Simply write out all the loop and node rules for a circuit. Then place coefficients of the equations you created in a matrix in the format below. The matrix should be an augmented matrix with the elements on the left being the resistances and the right being the EMF of that loop.

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& & & | EMF \\
& & & | EMF
\end{bmatrix}.</math>

To row reduce you should add, subtract, multiply, interchange or add a multiple of rows to each other until the matrix is as far simplified as possible. This is mostly related to concepts from linear algebra. A good way to think about is algebraic manipulation: the goal is get simple terms in the matrix through normal mathematical manipulations.

===A Computational Model===

Click here for a computational model of an [http://www.falstad.com/circuit/ online circuit simulator]. It opens up with an LRC circuit that has current running through it. The graphs on the bottom show the voltage as the current runs through the circuit. You can see that after a full loop the voltage is 0, verifying the loop rule. The voltage is always conserved, as long as the node rule can apply.

Another computational model, taken from an open source article [https://rhettallain.com/2019/06/21/rc-circuit-as-an-example-of-the-loop-rule/ is given here]. This depicts the loop rule in a RC circuit with a bulb. It depicts how the net voltage in the bulb tends to zero over time, thus signifying a physical example of the loop rule in action. The Glowscript code can be found [https://www.glowscript.org/#/user/achaudhury9/folder/MyPrograms/program/LoopRuleRC here]. The result is shown below:

[[File: Screenshot_2021-11-28_at_21-56-49_GlowScript_IDE.png ‎]]

==Examples==

===Simple===
'''Question'''

[[File: Node_comp.gif|thumb|250 px|Figure 1]]

Figure 1 displays a node in a circuit. I1 is equal to 10 amps. I2 is equal to 4 amps. What is I3?

'''Solution'''

: The current flowing into the node: <math> I_1 = 10A </math> The current flowing out of the node: <math> I_2 + I_3 </math> We know that the current flowing in must equal the current flowing out, so <math>10A = 4A + I_3 </math> Therefore <math>I_3</math> must equal 6A.

'''Question'''

[[File:SimpleLoopRule.jpg|300 px]]

The circuit shown above consists of a single battery and a single resistor. The resistance of the wires is negligible for this problem.

If the <math> emf </math> is 5 V and the resistance of the resistor is 10 ohms, what is the current passing through the resistor?

'''Solution'''

Although we can solve this using the <math>V = IR</math> equation for the whole loop, let's examine this problem using the loop rule equation.

The loop rule equation would be <math> {V}_{battery} - {V}_{resistor} = 0 </math>

Since we know the <math> emf </math> of the battery we just need to find the potential difference through the resistor. For this we can use the equation of <math> V = IR </math>.

Thus we now have an loop rule equation of <math> emf - IR = 0 </math>
From here it is a relatively simple process to find the current. We can rewrite the loop rule equation as <math> emf = IR </math> and then plug in 5 for the emf and
10 for the resistance, leaving us with I = .5 amperes.

===Middling===

'''Question'''

[[File:ExampleLoopRule2.png|300 px]]

The circuit shown above consists of a single battery, whose <math> emf </math> is 1.3 V, and three wires made of the same material, but having different cross-sectional areas.
Let the length of the thin wires be <math> {L}_{thick} </math> and the length of the thin wire be <math> {L}_{thin} </math>
Find a loop rule equation that starts at the negative end of the battery and goes counterclockwise through the circuit.

'''Solution'''

When beginning this problem, you must notice that the difference in cross-sectional areas affects the electric field in each wire. Because of this
we will denote the electric field at D. as <math> {E}_{D} </math> and the electric field everywhere else as <math> {E}_{A} </math>.
To begin we will go around the circuit clockwise and add up each component. First, we know that the <math> emf </math> of the battery is 1.3 V. Then, we will add up the potential
voltage of each of the wires.

Remember that the electric potential of a wire is equal to the product of the electric field and the length of the wire. From this we can now find the potential difference
of each section of the wires. The electric potential of location A-C is <math> {E}_{A} * {L}_{thick} </math>. This is the same for the electric potential of locations E - G of the wire.
For the thin section of the wire, the electric potential is <math> {E}_{D} * {L}_{thin} </math>. From here we just go around the circuit counterclockwise and add each potential difference to the loop rule equation.

Thus we can find that a loop rule equation is: <math> emf - 2 ({E}_{A} * {L}_{thick}) - {E}_{D} * {L}_{thin} = 0 </math>

This can also be rewritten as: <math> emf = 2 ({E}_{A} * {L}_{thick}) + {E}_{D} * {L}_{thin} = 0 </math>

===Difficult===
'''Question'''

[[File: Middlenode.gif|thumb|250 px|Figure 2]]

In Figure 2, I1 equal 23 amps, I2 equals 5 amps and I3 equals 42 amps. What is I4?

'''Solution'''

:The current flowing into the node: <math> I_1 + I_2 = 23A + 5A = 28A </math> The current flowing out of the node: <math> I_3 + I_4 = 42A + I_4 </math> Applying the node rule by using substitution, <math> I_4 = -14A </math> But how could we get a negative current? This negative current implies that <math> I_4 </math> flows in the opposite direction of what we assumed. A negative current in a loop rule problem implies that a current is in the opposite direction.

'''Question'''

[[File:HardLoopRule.jpg|350 px]]

For the circuit above, imagine a situation where the switch has been closed for a long time.
Calculate the current at a,b,c,d,e and charge Q of the capacitor. Answer these using <math> emf, {R}_{1}, {R}_{2},
and \space C </math>

'''Solution'''

First, write loop rule equations for each of the possible loops in the circuit. There are 3 loop equations that are possible.

<math> emf - {I}_{1}{R}_{1} - {I}_{2}{R}_{2} = 0 </math>

<math> emf - {I}_{1}{R}_{1} - Q/C = 0 </math>

<math> {I}_{2}{R}_{2} - Q/C = 0 </math>

From here, we can then solve for the current passing through a,b,d and e. We also know that the current passing through
these points must be the same so <math> {I}_{1} = {I}_{2} </math>

<math> emf - {I}_{1}{R}_{1} - {I}_{1}{R}_{2} = 0 </math>

<math> emf = {I}_{1}({R}_{1} + {R}_{2}) = 0 </math>

<math> emf/({R}_{1} + {R}_{2}) = {I}_{1} </math>

So the current at <math>a,b,d,e = emf/({R}_{1} + {R}_{2}) </math>

You must also know that once a capacitor is charging for a long time, current no longer flows through the capacitor. We can then
easily solve for c because since current is no longer flowing through the capacitor, the current at c = 0.

We will use the loop rule equation of <math> {I}_{2}{R}_{2} - Q/C = 0 </math> to solve for Q.

<math> {I}_{2}{R}_{2} = Q/C </math>

<math> C*({I}_{2}{R}_{2}) = Q </math> Since <math> {I}_{1} = {I}_{2} </math>

<math> C*({I}_{1}{R}_{2}) = Q </math>

We will plug in what we found {I}_{1} equals from before.

<math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}) = Q </math>

Lastly, <math>Q = C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math> and we have now solved the problem.

Current at <math> a,b,d,e = emf/({R}_{1} + {R}_{2}) </math>

Current at <math> c = 0 </math>

<math> Q = C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math>

'''Question (USING MATRICES)'''

[[File:Phys_Wiki_Image_1.png]]

Six identical resistors and two different batteries are connected in a circuit as shown in the diagram. One of the batteries has a potential difference of 20 V and the other has a potential difference of 5 V. The direction of the currents in the circuit are indicated and labeled. Solve your loop and node equations to determine the current through resistorR6. The resistance of each resistor is 100 Ω.

'''Solution'''

From the diagram, it is evident that this particular problem will require quite a few loop and node rule equations, 11 total to be exact. To figure out the current through resistor R6 we will need to set up the node and loop rule equations in order to have a system of equations that will have enough variables and coefficients in order to be able to solve for this current.

The node equations are:

0 = I1 - I3 - I6

0 = I1 - I2 - I4

0 = I2 - I3 - I5

0 = I4 + I5 - I6

The loop rule equations are:

0 = -emf1 + (R1 + R4)I1 + (R5)I4 + (R4)I6

0 = -emf1 + (R1 + R6)I1 + (R2)I2 + (R3)I3

0 = -emf1 + (R1 + R6)I1 + (R2)I2 + emf2 + (R4)I6

0 = (R2)I2 + (R3)I3 - (R4)I6 - (R5)I4

0 = (R2)I2 - emf2 - (R5)I4

0 = (R3)I3 - (R4)I6 + emf2

0 = -emf1 + (R1 + R6)I1 + (R5)I4 + emf2 + (R3)I3

Now that we have the equations instead of taking the time to solve the system of equations for all 11 of these equations, we can simply use a matrix and row reduce in order to find the current is going through R6 as we should have enough relationships to deduce this. The matrix for this particular problem would be the following:

[[File:Phys_Wiki_Matrix.png]]

In the matrix above, the headers at the top represent what the coefficients in the matrix pertain too, being the emf and the resistances associated with each of the currents form I1 through I6. Now using a row reduction calculator we can make an augmented matrix with the results being the emf column and the currents and associated resistances being on the LHS and we can find the I in the right hand column of the augmented matrix that is associated with the correct part of the circuit we are looking for and that will be our result for the current through the circuit at that resistor. In this case, that ends up being 1/15 amps.

In this particular problem, using traditional system of equation methods would prove problematic and time consuming as there are 11 equations that we have to work with. Thus, by using a matrix and row reduction, we can simply place the coefficients into a matrix, row reduce, and look at the number that is in the column of the resistor we are trying to find the current for and that will be our solution.

==Connectedness==

The Node Rule is connected to a lot of other topics in physics. The loop rule is the most important one, as the node rule and loop rule in conjunction allow us to solve circuits. The node rule is also connected to other concepts such as voltage, current and electricity. The Node Rule also supports the law of conservation of energy because in essence, current is simply the flow of electric charge, and since you cannot (at least as of now) create energy or electrons out of nothing, everything that is put into the system must come out somehow. Therefore, all the current that is applied, must come out the other end. Kirchoff's laws preceeded Maxwell's, and played a key part in his equations.

The Loop Rule is simply an extension of the conservation of energy applied to circuits. Circuits are ubiquitous as they are featured in almost every technology today. Our understanding of these technologies is rooted in the empirical discoveries made in the mid 19th century, and further boosted by the the advancement of theoretical knowledge due to the likes of Faraday and Maxwell.

Perhaps the most interesting thing about Kirchoff's rules is that they can be derived from the fundamental laws of physics, and thus do not be explicitly memorized. They are simply extensions of the energy concepts we already know to be true.

One interesting note about Loop Rule is that is does not apply universally to all circuits. In particular, AC (alternating current) circuits at high frequencies, have a fluctuating
electric charge that changes direction. This causes the electric potential of a round trip path around the circuit to no longer be zero. However, DC (direct current)
circuits, and low frequency circuits in general, still follow the loop rule.

These laws allow for voltmeters and ammeters to work, with voltmeters having a high resistance and being connected in parallel and ammeters having a low resistance and being connected in series.

Linear algebra is also intertwined with Kirchhoff's laws involving loops and nodes as complex and large circuits with many components can often lead to many different loops and the amount of loop and node equations can often get out of hand very quickly. As a result, we can use matrices to simplify the algebra involved with solving for currents in components that are a part of large circuits with many different loop and node equations. After creating the matrix we need skills from linear algebra in order to find the row reduced form of the augmented matrix that we create. With this reduced version of the matrix, we can find the current or resistances of the different parts of the circuit in a much easier and more timely fashion.

==Applications==

===Circuit Design ===
Kirchhoff's Laws are vital when designing electrical circuits as well as when we are analyzing them. The circuits that these laws can help us design and analyze can range from those found in our household appliances to more complex ones used in industrial applications at a large scale. In engineering, these laws are relevant because they help us design circuits that meet specific performance requirements.

===Current Measurement===
Kirchhoff's Laws can be used to measure the current flowing at various specific points of a circuit. By applying these laws at specific locations in a circuit, it can help us determine the amount of current entering and leaving that location or node. This aids us when we want to measure and control the flow of current in a circuit while designing or analyzing said circuit.

===Voltage Measurement===
Kirchhoff's Laws can be used to measure the voltage at different points of a circuit. By applying these laws to a loop in a circuit, it can help us determine the amount of voltage across all the points in that loop. This aids us when we want to measure and control the voltage levels in a circuit while designing or analyzing said circuit.

==History==
[[File:KirchoffLoopRule.jpg|thumb|Gustav Kirchhoff]]

Gustav Kirchhoff was a German physicist who lived during the 19th century. There are many equations and laws named after him that he helped to discover. His circuit laws (the node rule and loop rule) were the first laws that he conceived, and discovered this during his time as a student at Albertus University of Königsberg in 1845; he later wrote his doctoral dissertation on these laws. Kirchoff went on to explore the topics of spectroscopy and black body radiation after his graduation from Albertus. In addition to his circuit laws, he is also known for his law of thermochemistry and three laws of spectroscopy, the latter of which helped lead to quantum mechanics.

== See Also ==

===Further Reading===

*[[Components]]

*[[Circular Loop of Wire]]

*[[Resistors and Conductivity]]

*[[Current]]

===External Links===

https://www.boundless.com/physics/textbooks/boundless-physics-textbook/circuits-and-direct-currents-20/kirchhoff-s-rules-152/the-junction-rule-539-6331/

https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s

http://www.tutorvista.com/content/physics/physics-iv/current-electricity/kirchhoffs-rules.php

http://physics.bu.edu/~duffy/py106/Kirchoff.html

https://www.mathportal.org/algebra/solving-system-of-linear-equations/row-reduction-method.php

==References==

*Matter & Interactions 4th Edition by Ruth W. Chabay & Bruce A. Sherwood
*http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=roc
*http://www.regentsprep.org/Regents/physics/phys03/bkirchof1/
*https://www.mathportal.org/algebra/solving-system-of-linear-equations/row-reduction-method.php
*https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s
* https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law
* https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-dc-circuit-analysis
* http://www.ux1.eiu.edu/~cfadd/1360/28DC/Loop.html
* http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html
* http://www.falstad.com/circuit/
* Schuster, D. (Producer). (2013). Kirchhoff's Loop and Junction Rules Theory [Motion picture]. United States of America: YouTube.
* Schuster, D. (Producer). (2013). Kirchhoff's Rules (Laws) Worked Example [Motion picture]. United States of America: YouTube.
* Anderson, P. (Producer). (2015). Kirchoff's Loop Rule [Motion picture]. United States of America: YouTube.

[[Category: Electric Fields and energy in circuits]]

Steady State

2023-11-11T16:24:51Z

Daniel72: /* Middling */ The equation & solution for the cross-sectional area of the wire were incorrect. Even though the correct value was used later.

'''Claimed Alex Dulisse (Spring 2018)'''

Imagine a circuit in which the only battery has been disconnected. In such a setup there would be no electric field in the circuit and therefore no moving charges so the circuit would be in static equilibrium. When the battery is reconnected, an electric field will quickly propagate through the circuit and the charges in the circuit will accelerate. After a short time, the charges will move at a constant speed through the circuit and the electric field at each part of the circuit will remain unchanged. This means that at any given moment, all properties of the circuit remain unchanged even though an electric field is acting on the charges. At this time, when the state of the circuit does not change with the passage of time, the circuit is said to be in '''steady state.'''

[[Image:current_thing.png|thumb|upright = 0.2|right|450px|Current Diagram]]

==The Main Idea==

If a circuit is in steady state the following will be true:
:* At each part of the circuit, the drift velocity of charges remains unchanged with time.
:* The maximum amount of energy has been stored in capacitors, inductors.
:* Excess charge only accumulates on the surface of the wire.
:* An unchanging voltage exists between any two points in the system.

Each of these bullets is a different way of saying the same thing, that the state of the circuitry acting on any charges remains unchanged with respect to time. It is important to remember that a circuit being in steady state does not mean that the drift speed is the same every where in the circuit, only that it is unchanging for each specific location in the circuit. Also, a circuit in which there is no current may still be in steady state (such as an RC circuit in which the capacitors are fully charged), as long as there is still a constant electric field in all conductive parts of the circuit.

'''A Conceptual Understanding'''

For charges to move through the circuit, there must be an applied electric field that causes the mobile charges to move. Since there cannot be excess charge in the interior of a conductor, the surface charges must be producing the electric field. This electric field must be parallel to the wire because of the properties of a conductor. In a conductor, mobile charges always move to cancel out an electric field, meaning that the only electric field in the circuit must be parallel to the direction of current flow.
It is also important to conceptually understand why this electric field eventually reaches steady state. When a battery is first connected to a circuit, the electric field increases in magnitude, the rate at which the charges move through the circuit increases in magnitude too. As charges move through the circuit, there is a drag force in the opposite direction of their motion (similar to friction) known as the resistance of the wire. Since <math>I=V/R</math>, eventually these opposing forces will be equal in magnitude, bringing the circuit to steady state. Once a circuit is described as being in the steady state, there must be a constant electric field in the wire, the electric field has uniform magnitude throughout wire sections with similar cross-sectional area, and the electric field is parallel to the wire at every location along the wire.

[[File:Wire_Charges2.png|thumb = Wire_Charges2.png|upright=0.1|400px|center|Surface Charge Distribution]]

'''Steady State vs. Other States'''
When circuits move from static equilibrium to steady state, they typically do so in a very small amount of time. However, the [[Non Steady State]], or Transient State, which takes place in between, is an important aspect of circuits. When a circuit is in the Non Steady State, over time its state will approach the steady state. After enough time has elapsed, the difference is so slim that the circuit can be considered to be in steady state, even though in theory it never quite reaches steady state. The equations to model Non Steady States are different depending on the type of circuit being analyzed.

The following chart compares common properties of circuits in Static Equilibrium, Steady State, and Non Steady States:

{| class="wikitable"
|-
!
! Static Equilibrium
! Steady State
! Non Steady State
|-
| <math>\bar{v}</math>
| <math>= 0</math>
| <math>\neq 0</math>
| <math>\neq 0</math>
|-
| <math>\frac{d\bar{v}}{dt}</math>
| <math>= 0</math>
| <math>= 0</math>
| <math>\neq 0</math>
|-
| <math>\sum E_{inside}</math>
| <math>= 0</math>
| <math>\neq 0</math>
| <math>\neq 0</math>
|-
|}

===A Mathematical Model===
'''Current'''

There are two ways to describe current in a circuit, the electron current (<math>i</math>), and the conventional current (<math>I</math>). Electron current describes the motion of charges as it actually happens in the circuit: electrons moving through the conductor. Conventional current has the same magnitude and opposite direction as electron current as it describes positive charges moving through a circuit. Although circuits will sometimes act like positive charges are moving through the circuit if there are “electron holes,” it is important to understand that there are not positively charged particles moving through circuits. Even though it is backwards from the physical reality, conventional current is more commonly used because the motion of electrons in circuits was not understood when the first discoveries of electricity were made.
The following formulas can be used to describe electron and conventional current, respectively:
 <math>i= nA\bar{v}</math>

<math>I = \left | q \right |nA\bar{v}</math>

n = charge density, the number of charged particles per unit volume

A = cross sectional area of the wire

v = drift speed, the speed at which mobile charges move through a section of wire

q = charge of the mobile particles being described (only matters for conventional current)

Keep in mind that drift velocity <math>\bar{v}=\mu \left | E_{applied} \right |</math>

Where <math> \mu </math> = electron mobility.

If you have trouble thinking of this conceptually, imagine a pipe carrying water. For a certain amount of water to move through each part of the pipe per unit time, if the pipe is smaller, the water must move faster and for more water to be delivered, either the speed of the water must increase or the size of the pipe must increase.

'''Kirchoff's Node Rule'''

When a wire splits in to two, this junction is referred to as a node. Current flowing through the node must follow the equation <math>\Sigma I_{in} = \Sigma I_{out}</math>.
This is called the “[[Node Rule]].” All this is saying is that exactly what enters the node must exit. In other words, charge dose not disappear or come into existence just because one wire splits in two.

[[Image:NodeRule2.jpg|thumbnail = NodeRule2.jpg|upright=0.5|400 px|center|Node Rule]]

==Examples==

===Simple===
'''Question'''

What are the values of X and Y in the following circuit?
[[Image:Node_rule_simple_problem.png|thumbnail = Node_rule_simple_problem.png|upright=0.5|frame|center|Node Rule]]

'''Solution'''

Based on the node rule, 10=3+3+X. Therefore X=4. 
Y=3+X, therefore Y=3+4=7 

===Middling===

[http://cnx.org/contents/Ax2o07Ul@9.39:3ct4v3c5@4/Current| (Example taken from the OpenStax College Physics Textbook)]

'''Question'''

Calculate the drift velocity of electrons in a 12-gauge copper wire (which has a diameter of '''2.053 mm''') carrying a '''20.0-A''' current, given that there is one free electron per copper atom. The density of copper is <math>8.80\times 10^{3} kg/m^3</math>.

'''Solution'''

First, calculate the density of free electrons in copper. There is one free electron per copper atom. Therefore, is the same as the number of copper atoms per <math>m^3</math>. We can now find <math>n</math> as follows:

<math>n= \frac{1e-}{atom}\times \frac{6.02\times10^{23}atoms}{mol}\times \frac{1 mol}{63.54 g} \times \frac{1000 g}{kg} \times \frac{8.80 \times 10^3 kg}{1 m}</math>

 <math>n= 8.342\times 10^{28}e−/m^3</math>
 The cross-sectional area of the wire is
<math>A= \pi r^2 = \pi(2.053\times0.5\times10^{−3})^2 = 3.3103 \times 10^{–6}m^2</math>

 Rearranging <math>I= \left | q \right |nA\bar{v}</math> to isolate drift velocity gives you:
 <math>\bar{v}= \frac{I}{n\left |q \right |A}</math>
 <math>\bar{v}= \frac{20.0A}{(8.342\times10^{28}/m^3) (1.60\times10^{-19}C)(3.310\times10^{-6}m^2)}</math>
 <math>\bar{v}= 4.53\times 10^{–4}m/s </math>

===Difficult===

'''Question'''

A circuit made of two wires is in the steady state. The battery has an emf of '''1.5 V'''. Both wires are length '''25 cm''' and made of the same material. The thin wire's diameter is '''0.25 mm''', and the thick wire's diameter is '''0.31 mm'''. There are <math>4\times 10^{28}</math> mobile electrons per cubic meter of this material, and the electron mobility of this material is '''0.0006 (m/s)/(V/m)'''.

A) If the magnitude of the electric field in the thin wire is '''3.5 V/m''', what is the electric field in the thick wire? 
B) What is the drift velocity in the thin and thick wires?

[[File:sscircuit2.png]]

'''Solution'''
 Based on the Node Rule, we know that the electric current is equal in both wires:
 <math>i_{thick}=i_{thin}</math>
 We also know that <math>i= nA\bar{v} = nA \mu E</math>. Therefore:
 <math>nA_{thick} \mu E_{thick} = nA_{thin} \mu E_{thin}</math>
 When we rearrange this to solve for <math>E_{thick}</math>, the electron mobility (<math> \mu </math>) and electron density (<math>n</math>) cancel out and we are left with:
 <math>E_{thick}= \frac{A_{thin}E_{thin}}{A_{thick}}</math>

 The cross-sectional area of the thin wire is <math>A_{thin}= \pi (d_{thin}/2)^2 = \pi(0.125\times10^{−3}m^2) = 4.91 \times 10^{–8}m^2.</math>
 The cross-sectional area of the thick wire is <math>A_{thick}= \pi (d_{thick}/2)^2 = \pi(0.155\times10^{−3}m^2) = 7.55 \times 10^{–8}m^2.</math>

 <math>E_{thick}= \frac{(4.91 \times 10^{–8}m^2)(3.5 V/m)}{7.55 \times 10^{–8}m^2}</math>
 <math>E_{thick}=2.28 V/m</math>

 We know that <math>\bar{v}=\mu \left | E_{applied} \right |</math>.
 The drift velocity of the thin wire is <math>\bar{v}=\mu E_{thin} = [0.0006 (m/s)/(V/m)][3.5 V/m]= 2.10 \times 10^{–3} m/s</math>.
 The drift velocity of the thick wire is <math>\bar{v}=\mu E_{thick} = [0.0006 (m/s)/(V/m)][2.28 V/m]= 1.37 \times 10^{–3} m/s</math>.

==Connectedness==
As an industrial engineer, I often do not use physics. However, when designing efficient systems, concepts similar to those of a circuit appear. For instance, if a number of people need to get from one floor of a building down to another floor and they have a large staircase and a small staircase connecting the two floors, it is most efficient for them to move like electrons would move in a parallel circuit with a strong and weak resistor. It makes sense that these similarities would exist because circuits are essentially charges moving from point A to point B while expending the least amount of energy possible. That being said, in specific circumstances understanding physics concepts can be essential to even the work of an industrial engineer, especially in laying out factories to use the least amount of energy.

== See also ==
===Further Reading===
* [[Current]]
* [[Fundamentals of Resistance]]
* [[Node Rule]]
* [[Loop Rule]]
* [[Ohm's Law]]
* [[RL Circuit]]
* [[Series Circuits]]
* [[Parallel Circuits]]
===External Links===
* https://www.youtube.com/watch?v=Byvz0MMH_fw
* https://www.amherst.edu/system/files/media/1182/Lecture15%252520slides.pdf

==References==

*[https://openstax.org/details/college-physics| OpenStax College Physics]
*[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html| HyperPhysics]
*[https://www.codecogs.com/latex/eqneditor.php Formula code created with Online LaTeX Equation Editor]
*[https://en.wikipedia.org/wiki/Steady_state]

Kirchoff's Laws

2023-11-11T16:17:32Z

Daniel72: Swapped the order of how the first law is introduced to make it consistent with how the second law is introduced.

Edits: Regina Ivanna Gomez Quiroz, Spring 2023

[[File:Junction.gif|thumb|The node (or junction) rule states that the current flowing in is equal to the current that flows out.
I1 + I4 = I2 + I3 + I5]]

Kirchoff's Laws are two fundamental principles of electric circuits and are used to determine the behaviors of electric circuits and their components. They serve as a guide for how circuits will behave and are accurate for all DC and low-frequency AC circuits. These principles are used to measure voltage and current in [http://www.physicsbook.gatech.edu/Ammeters,Voltmeters,Ohmmeters voltmeters and ammeters].

Kirchoff's First Law, also known as Kirchoff's Node Rule or Kirchoff's Junction Rule, further exercises the law of Conservation of Charge and states that if current is constant, all the current that flows through one junction must be equal to all the current that flows out of the junction. This rule can be applied both to [[Electron and Conventional Current|conventional]] and [[Electron and Conventional Current|electron currents]]. This rule is not a fundamental principle, but rather a consequence of the fundamental principle of conservation of charge and the definition of steady-state. This law essentially states that the net amount of current entering a particular juncture in the system is equal to the net amount of current leaving the system. A mathematical way to think about is that the sum of the currents entering a point will be equal to the sum leaving it.

[[File:Visual_Model2.png|thumb|The Loop rule states that the sum of voltage around a loop is equal to 0.
LOOP 1: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CF} + \Delta {V}_{FA} = 0 </math>

LOOP 2: <math> \Delta {V}_{FC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} = 0 </math>

LOOP 3: <math> \Delta {V}_{AB} + \Delta {V}_{BC} + \Delta {V}_{CD} + \Delta {V}_{DE} + \Delta {V}_{EF} + \Delta {V}_{FA} = 0 </math>]]

Kirchoff's Second Law, also known as Kirchhoff's Loop Rule or Kirchhoff's Voltage Law states that the sum of potential differences around a closed circuit is equal to zero. More simply, in a completed circuit, the voltages around a loop will sum to 0. This is because voltage is just energy per unit charge, and both energy and charge are conserved by fundamental laws. A good way to think about this is by attaching a positive value to the points in the loop that contribute to the net voltage, and by attaching a negative value to the resistors. The amount of voltage that enters the loop will be equal to the amount that is dissipated across the loop, thus ensuring that the net voltage is zero.

Note that this is only true when the magnetic field is neither fluctuating nor time-varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.

Both laws have limitations; most specifically, the node rule relies on the net charge in the system being constant, while loop rule depends on time-varying magnetic fields being confined to singular parts.

==The Main Idea==

'''Kirchoff's Node Rule'''

The node rule states that at any junction in an electrical circuit, the amount of current flowing into the junction is equal to the amount of current flowing out of the junction in steady-state.

In the steady state, for many electrons flowing into and out of a node,
* electron current: <math> net\ i_{in} = net\ i_{out},</math> where <math>i = nA\mu</math><math>E</math>
* conventional current: <math> net\ I_{in} = net\ I_{out},</math> where <math>I = |q|nA\mu</math><math>E</math>

'''Conservation of Charge'''

This rule is an application of the conservation of electric charge, basically that charge within a circuit cannot be created or lost. During the flow process around the circuit, there is no loss of any charge, thus the total current in any cross-section of the circuit is the same. If there are no nodes in the loop, the conventional current is the same throughout the loop. This also why the node rule relies on a constant charge throughout the system; if the charge was constantly changing, then the node rule will not apply. The node rule essentially states that the net current entering a juncture in the system will always be constant.

'''Kirchoff's Loop Rule'''

The loop rule simply states that in any round trip path in a circuit,
[[Electric Potential]] equals zero. This applies through any round trip path; in more complex circuits, there can be
multiple round trip paths. This principle is an application of the conservation of energy, specifically within a circuit.
This principle is often used to solve for resistance or current passing through of light bulbs and other resistors, as well as the capacitance or charge of capacitors in a circuit.

[[File:loopexample.png]]

LOOP 1: <math> \Delta {V}_1 = emf - I_1R_1 = 0 </math>

LOOP 2: <math> \Delta {V}_2 = -I_1R_1 + I_2R_2 = 0 </math>

LOOP 3: <math> \Delta {V}_3 = emf - I_2R_2 = 0 </math>

To figure out the sign of the voltages, act as an observer walking along the path. Start at the negative end of the emf and continue walking along the path. The emf will be positive in the loop rule because you are moving from low to high voltage. Once you reach a resistor or capacitor, this will be negative in the loop rule equation because it is high to low voltage. Continue along the path until you return to the starting position.

'''Bringing Both Laws Together'''

Find all possible node and loop equations for the following circuit.

[[File:Non_steady_state_image_TWO.png|400 px]]

You should have 3 Loop Equations and 2 Node equations.

Some examples:

<math> I_1= I_2+I_3 </math>

<math> I_1*R_1 + emf_1 + emf_3 + I_3*R_4 = 0 </math>

'''Matrices in Conjunction with Kirchhoff's Laws'''

After creating loop equations for all possible loops in a circuit using the node rule and loop rule, you can input the coefficients of the equations into a matrix. After inputting it into the matrix, you can use linear algebra and row reduction in order to solve for the current or resistances of the different parts of the circuit. This format and method of understanding complex circuits are especially useful for circuits with a large number of loops as it makes the algebra involved in getting the currents much simpler and straightforward. Simply creating the equations for every loop possible and every node and then inputting the coefficients into a row reduction calculator can give you all the values for I and/or R that you might need.

After finding the loop equations you can use a matrix calculator as well to further simplify your task!: [http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=rref Online Matrix Row Reduction Calculator]

An example of how you would format the matrix of a Kirchhoff's law problem is shown below:

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& & & | EMF \\
& & & | EMF
\end{bmatrix}.</math>

Simply place to coefficients or resistances of each of the different resistors as separate elements in the columns of the matrix while keeping track of which resistance pertains to which part of the loop and place the EMF of that loop as the solution column in the augmented matrix. Then after row reducing, each of the resistors should be simplified so that there is a 1 in at least 1 row for every resistor and the current at the end of that row is the current for the resistor that is in the same column as the 1 in the row of the current.

For example in the matrix below the current going through R2 is 1/15 amps:

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& 1 & & | 1/15 \\
& & & | EMF
\end{bmatrix}.</math>

'''Limitations'''

The Node rule assumes that current flows only in conductors and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for high-frequency AC circuits. In other words, the node is valid only if the total electric charge, <math>Q</math>, remains constant in the region being considered. In practical cases, this is always so when the node rule is applied at a point.

The Loop rule is based on the assumption that there is no fluctuating magnetic field linking the closed-loop.
This is not a safe assumption for high-frequency AC circuits. In the presence of a changing magnetic field, the electric field is not a conservative vector field. Therefore, the electric field cannot be the gradient of any potential. That is to say, the line integral of the electric field around the loop is not zero, directly contradicting KVL.

Understanding these limitations is essential to understanding how motional emf, motors, and generators work.

===A Mathematical Model===

'''Kirchoff's Node Rule'''

[[File:kircho2.gif|right]]
The node rule can be stated as:
:<math> \sum \Delta{I} = 0 </math> 
where <math>I</math> stands for the current of the individual parts or wires in a circuit and the sign of the current that flows into the junction is opposite of the current that flows out of the junction. This simply supports the idea that charge cannot be created or destroyed because the total current must remain equal regardless of the path it takes.
:<math> \sum_{k=1}^n I_k = 0 </math> 
where <math>n</math> is the total number of branches with current flowing through the node, as well as along any node in a circuit:

:<math> \Delta {I}_{1} + \Delta {I}_{2} + \space.... = 0 </math>

or more generally:

:<math> \sum_{k=1}^n \tilde{I_k}</math> = 0 

Another definition is simply a summation rule:
"<math> \sum I(in) = \sum I(out)</math>
'''Kirchoff's Loop Rule'''

A mathematical representation is:

:<math> \sum \Delta{V} = 0 </math> 
where <math>V</math> stands for the voltage of the individual parts or wires in a circuit and the sign of the voltage that flows clock-wise is opposite of the current that flows counterclockwise. This simply supports the idea that energy cannot be created or destroyed because the total current must remain equal regardless of the path it takes.

:<math> \sum_{i=1}^n {V}_{i} = 0 </math>

where <math>n</math> is the number of voltages being measured in the loop, as well as

:<math> \Delta {V}_{1} + \Delta {V}_{2} + \space.... = 0 </math>

along any closed path in a circuit. The voltages may also be complex:

:<math>\sum_{k=1}^n \tilde{V}_k = 0</math>

'''Matrix Intrepretation'''

Using matrices can simplify the process of using node and loop rules in order to find out different parts of a complex circuit. Simply write out all the loop and node rules for a circuit. Then place coefficients of the equations you created in a matrix in the format below. The matrix should be an augmented matrix with the elements on the left being the resistances and the right being the EMF of that loop.

:<math>\mathbf{A} = \begin{bmatrix}
R1 & R2 & R3 & | EMF \\
& & & | EMF \\
& & & | EMF
\end{bmatrix}.</math>

To row reduce you should add, subtract, multiply, interchange or add a multiple of rows to each other until the matrix is as far simplified as possible. This is mostly related to concepts from linear algebra. A good way to think about is algebraic manipulation: the goal is get simple terms in the matrix through normal mathematical manipulations.

===A Computational Model===

Click here for a computational model of an [http://www.falstad.com/circuit/ online circuit simulator]. It opens up with an LRC circuit that has current running through it. The graphs on the bottom show the voltage as the current runs through the circuit. You can see that after a full loop the voltage is 0, verifying the loop rule. The voltage is always conserved, as long as the node rule can apply.

Another computational model, taken from an open source article [https://rhettallain.com/2019/06/21/rc-circuit-as-an-example-of-the-loop-rule/ is given here]. This depicts the loop rule in a RC circuit with a bulb. It depicts how the net voltage in the bulb tends to zero over time, thus signifying a physical example of the loop rule in action. The Glowscript code can be found [https://www.glowscript.org/#/user/achaudhury9/folder/MyPrograms/program/LoopRuleRC here]. The result is shown below:

[[File: Screenshot_2021-11-28_at_21-56-49_GlowScript_IDE.png ‎]]

==Examples==

===Simple===
'''Question'''

[[File: Node_comp.gif|thumb|250 px|Figure 1]]

Figure 1 displays a node in a circuit. I1 is equal to 10 amps. I2 is equal to 4 amps. What is I3?

'''Solution'''

: The current flowing into the node: <math> I_1 = 10A </math> The current flowing out of the node: <math> I_2 + I_3 </math> We know that the current flowing in must equal the current flowing out, so <math>10A = 4A + I_3 </math> Therefore <math>I_3</math> must equal 6A.

'''Question'''

[[File:SimpleLoopRule.jpg|300 px]]

The circuit shown above consists of a single battery and a single resistor. The resistance of the wires is negligible for this problem.

If the <math> emf </math> is 5 V and the resistance of the resistor is 10 ohms, what is the current passing through the resistor?

'''Solution'''

Although we can solve this using the <math>V = IR</math> equation for the whole loop, let's examine this problem using the loop rule equation.

The loop rule equation would be <math> {V}_{battery} - {V}_{resistor} = 0 </math>

Since we know the <math> emf </math> of the battery we just need to find the potential difference through the resistor. For this we can use the equation of <math> V = IR </math>.

Thus we now have an loop rule equation of <math> emf - IR = 0 </math>
From here it is a relatively simple process to find the current. We can rewrite the loop rule equation as <math> emf = IR </math> and then plug in 5 for the emf and
10 for the resistance, leaving us with I = .5 amperes.

===Middling===

'''Question'''

[[File:ExampleLoopRule2.png|300 px]]

The circuit shown above consists of a single battery, whose <math> emf </math> is 1.3 V, and three wires made of the same material, but having different cross-sectional areas.
Let the length of the thin wires be <math> {L}_{thick} </math> and the length of the thin wire be <math> {L}_{thin} </math>
Find a loop rule equation that starts at the negative end of the battery and goes counterclockwise through the circuit.

'''Solution'''

When beginning this problem, you must notice that the difference in cross-sectional areas affects the electric field in each wire. Because of this
we will denote the electric field at D. as <math> {E}_{D} </math> and the electric field everywhere else as <math> {E}_{A} </math>.
To begin we will go around the circuit clockwise and add up each component. First, we know that the <math> emf </math> of the battery is 1.3 V. Then, we will add up the potential
voltage of each of the wires.

Remember that the electric potential of a wire is equal to the product of the electric field and the length of the wire. From this we can now find the potential difference
of each section of the wires. The electric potential of location A-C is <math> {E}_{A} * {L}_{thick} </math>. This is the same for the electric potential of locations E - G of the wire.
For the thin section of the wire, the electric potential is <math> {E}_{D} * {L}_{thin} </math>. From here we just go around the circuit counterclockwise and add each potential difference to the loop rule equation.

Thus we can find that a loop rule equation is: <math> emf - 2 ({E}_{A} * {L}_{thick}) - {E}_{D} * {L}_{thin} = 0 </math>

This can also be rewritten as: <math> emf = 2 ({E}_{A} * {L}_{thick}) + {E}_{D} * {L}_{thin} = 0 </math>

===Difficult===
'''Question'''

[[File: Middlenode.gif|thumb|250 px|Figure 2]]

In Figure 2, I1 equal 23 amps, I2 equals 5 amps and I3 equals 42 amps. What is I4?

'''Solution'''

:The current flowing into the node: <math> I_1 + I_2 = 23A + 5A = 28A </math> The current flowing out of the node: <math> I_3 + I_4 = 42A + I_4 </math> Applying the node rule by using substitution, <math> I_4 = -14A </math> But how could we get a negative current? This negative current implies that <math> I_4 </math> flows in the opposite direction of what we assumed. A negative current in a loop rule problem implies that a current is in the opposite direction.

'''Question'''

[[File:HardLoopRule.jpg|350 px]]

For the circuit above, imagine a situation where the switch has been closed for a long time.
Calculate the current at a,b,c,d,e and charge Q of the capacitor. Answer these using <math> emf, {R}_{1}, {R}_{2},
and \space C </math>

'''Solution'''

First, write loop rule equations for each of the possible loops in the circuit. There are 3 loop equations that are possible.

<math> emf - {I}_{1}{R}_{1} - {I}_{2}{R}_{2} = 0 </math>

<math> emf - {I}_{1}{R}_{1} - Q/C = 0 </math>

<math> {I}_{2}{R}_{2} - Q/C = 0 </math>

From here, we can then solve for the current passing through a,b,d and e. We also know that the current passing through
these points must be the same so <math> {I}_{1} = {I}_{2} </math>

<math> emf - {I}_{1}{R}_{1} - {I}_{1}{R}_{2} = 0 </math>

<math> emf = {I}_{1}({R}_{1} + {R}_{2}) = 0 </math>

<math> emf/({R}_{1} + {R}_{2}) = {I}_{1} </math>

So the current at <math>a,b,d,e = emf/({R}_{1} + {R}_{2}) </math>

You must also know that once a capacitor is charging for a long time, current no longer flows through the capacitor. We can then
easily solve for c because since current is no longer flowing through the capacitor, the current at c = 0.

We will use the loop rule equation of <math> {I}_{2}{R}_{2} - Q/C = 0 </math> to solve for Q.

<math> {I}_{2}{R}_{2} = Q/C </math>

<math> C*({I}_{2}{R}_{2}) = Q </math> Since <math> {I}_{1} = {I}_{2} </math>

<math> C*({I}_{1}{R}_{2}) = Q </math>

We will plug in what we found {I}_{1} equals from before.

<math> C* (emf/({R}_{1} + {R}_{2})){R}_{2}) = Q </math>

Lastly, <math>Q = C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math> and we have now solved the problem.

Current at <math> a,b,d,e = emf/({R}_{1} + {R}_{2}) </math>

Current at <math> c = 0 </math>

<math> Q = C* (emf/({R}_{1} + {R}_{2})){R}_{2}</math>

'''Question (USING MATRICES)'''

[[File:Phys_Wiki_Image_1.png]]

Six identical resistors and two different batteries are connected in a circuit as shown in the diagram. One of the batteries has a potential difference of 20 V and the other has a potential difference of 5 V. The direction of the currents in the circuit are indicated and labeled. Solve your loop and node equations to determine the current through resistorR6. The resistance of each resistor is 100 Ω.

'''Solution'''

From the diagram, it is evident that this particular problem will require quite a few loop and node rule equations, 11 total to be exact. To figure out the current through resistor R6 we will need to set up the node and loop rule equations in order to have a system of equations that will have enough variables and coefficients in order to be able to solve for this current.

The node equations are:

0 = I1 - I3 - I6

0 = I1 - I2 - I4

0 = I2 - I3 - I5

0 = I4 + I5 - I6

The loop rule equations are:

0 = -emf1 + (R1 + R4)I1 + (R5)I4 + (R4)I6

0 = -emf1 + (R1 + R6)I1 + (R2)I2 + (R3)I3

0 = -emf1 + (R1 + R6)I1 + (R2)I2 + emf2 + (R4)I6

0 = (R2)I2 + (R3)I3 - (R4)I6 - (R5)I4

0 = (R2)I2 - emf2 - (R5)I4

0 = (R3)I3 - (R4)I6 + emf2

0 = -emf1 + (R1 + R6)I1 + (R5)I4 + emf2 + (R3)I3

Now that we have the equations instead of taking the time to solve the system of equations for all 11 of these equations, we can simply use a matrix and row reduce in order to find the current is going through R6 as we should have enough relationships to deduce this. The matrix for this particular problem would be the following:

[[File:Phys_Wiki_Matrix.png]]

In the matrix above, the headers at the top represent what the coefficients in the matrix pertain too, being the emf and the resistances associated with each of the currents form I1 through I6. Now using a row reduction calculator we can make an augmented matrix with the results being the emf column and the currents and associated resistances being on the LHS and we can find the I in the right hand column of the augmented matrix that is associated with the correct part of the circuit we are looking for and that will be our result for the current through the circuit at that resistor. In this case, that ends up being 1/15 amps.

In this particular problem, using traditional system of equation methods would prove problematic and time consuming as there are 11 equations that we have to work with. Thus, by using a matrix and row reduction, we can simply place the coefficients into a matrix, row reduce, and look at the number that is in the column of the resistor we are trying to find the current for and that will be our solution.

==Connectedness==

The Node Rule is connected to a lot of other topics in physics. The loop rule is the most important one, as the node rule and loop rule in conjunction allow us to solve circuits. The node rule is also connected to other concepts such as voltage, current and electricity. The Node Rule also supports the law of conservation of energy because in essence, current is simply the flow of electric charge, and since you cannot (at least as of now) create energy or electrons out of nothing, everything that is put into the system must come out somehow. Therefore, all the current that is applied, must come out the other end. Kirchoff's laws preceeded Maxwell's, and played a key part in his equations.

The Loop Rule is simply an extension of the conservation of energy applied to circuits. Circuits are ubiquitous as they are featured in almost every technology today. Our understanding of these technologies is rooted in the empirical discoveries made in the mid 19th century, and further boosted by the the advancement of theoretical knowledge due to the likes of Faraday and Maxwell.

Perhaps the most interesting thing about Kirchoff's rules is that they can be derived from the fundamental laws of physics, and thus do not be explicitly memorized. They are simply extensions of the energy concepts we already know to be true.

One interesting note about Loop Rule is that is does not apply universally to all circuits. In particular, AC (alternating current) circuits at high frequencies, have a fluctuating
electric charge that changes direction. This causes the electric potential of a round trip path around the circuit to no longer be zero. However, DC (direct current)
circuits, and low frequency circuits in general, still follow the loop rule.

These laws allow for voltmeters and ammeters to work, with voltmeters having a high resistance and being connected in parallel and ammeters having a low resistance and being connected in series.

Linear algebra is also intertwined with Kirchhoff's laws involving loops and nodes as complex and large circuits with many components can often lead to many different loops and the amount of loop and node equations can often get out of hand very quickly. As a result, we can use matrices to simplify the algebra involved with solving for currents in components that are a part of large circuits with many different loop and node equations. After creating the matrix we need skills from linear algebra in order to find the row reduced form of the augmented matrix that we create. With this reduced version of the matrix, we can find the current or resistances of the different parts of the circuit in a much easier and more timely fashion.

==Applications==

===Circuit Design ===
Kirchhoff's Laws are vital when designing electrical circuits as well as when we are analyzing them. The circuits that these laws can help us design and analyze can range from those found in our household appliances to more complex ones used in industrial applications at a large scale. In engineering, these laws are relevant because they help us design circuits that meet specific performance requirements.

===Current Measurement===
Kirchhoff's Laws can be used to measure the current flowing at various specific points of a circuit. By applying these laws at specific locations in a circuit, it can help us determine the amount of current entering and leaving that location or node. This aids us when we want to measure and control the flow of current in a circuit while designing or analyzing said circuit.

===Voltage Measurement===
Kirchhoff's Laws can be used to measure the voltage at different points of a circuit. By applying these laws to a loop in a circuit, it can help us determine the amount of voltage across all the points in that loop. This aids us when we want to measure and control the voltage levels in a circuit while designing or analyzing said circuit.

==History==
[[File:KirchoffLoopRule.jpg|thumb|Gustav Kirchhoff]]

Gustav Kirchhoff was a German physicist who lived during the 19th century. There are many equations and laws named after him that he helped to discover. His circuit laws (the node rule and loop rule) were the first laws that he conceived, and discovered this during his time as a student at Albertus University of Königsberg in 1845; he later wrote his doctoral dissertation on these laws. Kirchoff went on to explore the topics of spectroscopy and black body radiation after his graduation from Albertus. In addition to his circuit laws, he is also known for his law of thermochemistry and three laws of spectroscopy, the latter of which helped lead to quantum mechanics.

== See Also ==

===Further Reading===

*[[Components]]

*[[Circular Loop of Wire]]

*[[Resistors and Conductivity]]

*[[Current]]

===External Links===

https://www.boundless.com/physics/textbooks/boundless-physics-textbook/circuits-and-direct-currents-20/kirchhoff-s-rules-152/the-junction-rule-539-6331/

https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s

http://www.tutorvista.com/content/physics/physics-iv/current-electricity/kirchhoffs-rules.php

http://physics.bu.edu/~duffy/py106/Kirchoff.html

https://www.mathportal.org/algebra/solving-system-of-linear-equations/row-reduction-method.php

==References==

*Matter & Interactions 4th Edition by Ruth W. Chabay & Bruce A. Sherwood
*http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=roc
*http://www.regentsprep.org/Regents/physics/phys03/bkirchof1/
*https://www.mathportal.org/algebra/solving-system-of-linear-equations/row-reduction-method.php
*https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s
* https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s_voltage_law
* https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-dc-circuit-analysis
* http://www.ux1.eiu.edu/~cfadd/1360/28DC/Loop.html
* http://higheredbcs.wiley.com/legacy/college/cutnell/0470223553/concept_sims/sim34/sim34.html
* http://www.falstad.com/circuit/
* Schuster, D. (Producer). (2013). Kirchhoff's Loop and Junction Rules Theory [Motion picture]. United States of America: YouTube.
* Schuster, D. (Producer). (2013). Kirchhoff's Rules (Laws) Worked Example [Motion picture]. United States of America: YouTube.
* Anderson, P. (Producer). (2015). Kirchoff's Loop Rule [Motion picture]. United States of America: YouTube.

[[Category: Electric Fields and energy in circuits]]

Field of a Charged Rod

2023-10-11T15:29:00Z

Daniel72: Removed the other incorrect "double countings" of sqrt^{3/2}.

'''Michael Wise, Fall 2020'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. Often, when analyzing physical systems, it is the case that we're unable to analyze each of the individual particle that make up an object. Therefore, it is efficient to generalize collections of particles into shapes (in this case, a rod) whereby the mathematics corresponding to electric field calculations can be simplified. This can be done by adding up the contributions to the electric field made by parts of an object, approximating each part of an object as a point charge. A more accurate model, however, comes from adding up the contributions of infinitesimal pieces of the given object.

Some objects, such as rods, can be modeled as a uniformly charged object in order to calculate the electric field at some observation location. The following wiki page provides an overview of electric fields created by uniformly charged thin rods, briefly presenting its inception, some specific cases, and proof of concept experiments. The case of a uniformly charged thin rod is a fundamental example of electric field patterns and calculations within physics. Its implications can be applied to other charged objects, such as rings, disks, and spheres.

In practical situations, objects have charges spread all over their surface. When going about calculating the electric fields of these objects, we can either use one of two processes: numerical summation or integration. Technically speaking, both methods involve dividing an object into many pieces and summing the individual pieces' electric field contributions, but they are distinct. As with point charges, the direction of the field is determined by the sign of the object's charge (positive-points away, negative-points toward) and the magnitude of the field is determined by the observation distance and the magnitude of the object's charge.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Divide the charged object into small pieces. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. Choose an origin and the axes. Write an algebraic expression for the electric field <math>\Delta \vec{E}</math> due to one piece.

3. Add up the contributions of all pieces, either numerically or symbolically.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(x^2+y^2)^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(x^2+y^2)^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]

Field of a Charged Rod

2023-10-11T15:26:45Z

Daniel72: I revised what I believe was an error. The formula previously had \sqrt{term}^(3/2) when it should have been (term)^3/2 or \sqrt{term}^3. It was "double counting" the square root.

'''Michael Wise, Fall 2020'''

== The Main Idea ==

Previously, we've learned about the electric field of a point particle. Often, when analyzing physical systems, it is the case that we're unable to analyze each of the individual particle that make up an object. Therefore, it is efficient to generalize collections of particles into shapes (in this case, a rod) whereby the mathematics corresponding to electric field calculations can be simplified. This can be done by adding up the contributions to the electric field made by parts of an object, approximating each part of an object as a point charge. A more accurate model, however, comes from adding up the contributions of infinitesimal pieces of the given object.

Some objects, such as rods, can be modeled as a uniformly charged object in order to calculate the electric field at some observation location. The following wiki page provides an overview of electric fields created by uniformly charged thin rods, briefly presenting its inception, some specific cases, and proof of concept experiments. The case of a uniformly charged thin rod is a fundamental example of electric field patterns and calculations within physics. Its implications can be applied to other charged objects, such as rings, disks, and spheres.

In practical situations, objects have charges spread all over their surface. When going about calculating the electric fields of these objects, we can either use one of two processes: numerical summation or integration. Technically speaking, both methods involve dividing an object into many pieces and summing the individual pieces' electric field contributions, but they are distinct. As with point charges, the direction of the field is determined by the sign of the object's charge (positive-points away, negative-points toward) and the magnitude of the field is determined by the observation distance and the magnitude of the object's charge.

The process of finding the electric field due to charge distributed over an object has four steps:

1. Divide the charged object into small pieces. Make a diagram and draw the electric field <math>\Delta \vec{E}</math> contributed by one of the pieces.

2. Choose an origin and the axes. Write an algebraic expression for the electric field <math>\Delta \vec{E}</math> due to one piece.

3. Add up the contributions of all pieces, either numerically or symbolically.

4. Check that the result is physically correct.

=== A Mathematical Model ===
The process of calculating a uniformly charged rod's electric field is tedious, but breaking the process down into several steps makes the task at hand easier. Consider a uniformly charged thin rod of length <math>L</math> and positive charge <math>Q</math> centered on and lying along the x-axis. The rod is being observed from above at a point on the y-axis.

'''Step 1: Divide the Distribution into Pieces; Draw <math>\Delta \vec{E}</math>'''

Imagine dividing the rod into a series of very thin slices, each with the same charge <math>\Delta Q</math>. This charge <math>\Delta Q</math> is a small part of the overall charge. Picture it as a point charge. Each slice contributes its own electric field, <math>\Delta E</math>. Summing all these individual slices of <math>E</math> gives you the total electric field of the rod. This process approaches taking an integral, as each thickness approaches 0 and the the number of slices approaches infinity. Note that in this example, the variable that is changing for each slice is its x-coordinate.

'''Step 2: Write an Expression for the Electric Field Due to One Piece'''

The second step is to write a mathematical expression for the field <math>\Delta E</math> contributed by a single slice of the rod. We use the formula of the electric field for a point charge because we are imagining each slice as a point charge. First, determine <math>r</math>, the vector pointing from the source to the observation location. For our example, this is <math> r = obs - source = <0,y,0> - < x,0,0> = <-x,y,0></math>. Now use this to calculate the magnitude and direction of <math>r</math>. So <math>|\vec{r}| = \sqrt{(-x)^2 + y^2} = \sqrt{x^2 + y^2}</math> and <math>\hat{r} = \frac{\vec{r}}{\hat{r}} = \frac{< -x,y,0>}{\sqrt{x^2 + y^2}} </math>. <math> \hat{r}</math> is the vector portion of the expression for the field. The scalar portion is <math> \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{|\vec{r}|^2}</math>. Thus the expression for one slice of the rod is:
<math> \Delta \vec{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{\Delta Q}{(x^2+y^2)^{3/2}} \cdot < -x,y,0> </math>.

'''Determining <math>\Delta Q</math> and the integration variable'''

In the first step, we determined that the changing variable for this rod was its x-coordinate. This means the integration variable is <math> dx</math>. We need to put this integration variable into our expression for the electric field. More specifically, we need to express <math>\Delta Q</math> in terms of the integration variable. Recall that the rod is uniformly charged, so the charge on any single slice of it is: <math>
\Delta Q = (\frac{\Delta x}{L})\cdot Q</math>. This quantity can also be expressed in terms of the charge density.

'''Expression for <math> \Delta \vec{E}</math>

Substitute the expression for the integration variable into the formula for the electric field of one slice. Separating the equation into separate x and y components, we get <math> \Delta \vec{E_x} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{-x}{(\sqrt{x^2+y^2})^{3/2}} \cdot dx </math> and <math> \Delta \vec{E_y} = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(\sqrt{x^2+y^2})^{3/2}} \cdot dx </math>. Note that we have replaced <math> \Delta x </math> with <math> dx</math> in preparation for integration.

'''Step 3: Add Up the Contributions of All the Pieces'''

The third step is to sum all of our slices. We can go about this in two ways. One way is with numerical summation, or separating the object into a finite number of small pieces, calculating the individual contributing electric fields, and then summing them. Another, more precise method is to integrate. Most of the work of finding the field of a uniformly charged object is setting up this integral. If you have reached the correct expression to integrate, the rest is simple math. The bounds for integration are the coordinates of the start and stop of the rod. In this example the bounds are from <math>-L/2</math> to <math>+L/2</math>. So the expression is <math> \int\limits_{-L/2}^{L/2}\ \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{L} \cdot \frac{y}{(\sqrt{x^2+y^2})^{3/2}} \cdot dx. </math> Solving this gives the final expression <math> E_y = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{x} \cdot \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. Note that the field parallel to the x axis is zero. This can be observed due to the symmetry of the problem.
This equation can be written more generally as <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r} \cdot \frac{1}{(\sqrt{r^2+ (L/2)^2})} </math> where r represents the distance from the rod to the observation location.

'''Step 4:Checking the Result'''

Finally, the fourth step is to check the result. The units should be the same as the units of the expression for the electric field for a single point particle <math> E = \frac{1}{4\pi\epsilon_0} \cdot \frac{Q}{r^2} </math>.
Our answer has the right units, since <math> \frac{1}{(\sqrt{x^2+ (L/2)^2})} </math>. has the same units of <math>\frac{Q}{r^2}</math>

Is the direction qualitatively correct? We have the electric field pointing straight away from the midpoint of the rod, which is correct, given the symmetry of the situation. The vertical component of the electric field should indeed be zero.

=== Computational Models ===

'''Finding the Electric Field from a Rod with Code'''
While computation done by hand does have its merits, and certainly was the
methodology used when these ideas were conceived, there are much more efficient,
powerful, and most importantly, pretty ways to go about finding and showing the
electric field. This is of course referring to computers, specifically computers
running glowscript code in the case of Physics 2212. With that idea in mind,
here are some demonstrations of said methodology:

This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.

Notice the edge-effects of the electric field of the rod. For reasons
discussed above, if we used the long rod approximation (L>>d), these
effects would be negligible.

[http://www.glowscript.org/#/user/yoderlukas/folder/Public/program/ElectricFieldAlongRodLength '''Click Here to Run the Code''']

For a more precise model, the two links below lead to code that generates a
forty element line of charge with given magnitude, and length, and then
iterates vectors representing the electric field in the space all around said
line of charge. Note that the vectors are small, but for the positively charged
rod, they lead radially outward, and for the negative, radially inward. This is
due to the fact that the rod(s) are treated as lines of positive or negative
charge, and the electric field behaves as such.

Try zooming in and out! You can really see the symmetry of the field far out,
and the edge effects when zoomed in.

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Positive '''Positive Charge''']

[https://www.glowscript.org/#/user/michaelwise/folder/Public/program/LineofCharge-Negative '''Negative Charge''']

==Examples==

Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.

===Simple===

[[Image:LukasYoderNo.jpg|800px|center|thumb|Figure1: Problem 1]]

===Middling===

[[Image:LukasYoderMaybe.jpg|800px|center|thumb|Figure 2: Problem 2]]

===Difficult===

[[Image:LukasYoderYes.jpg|800px|center|thumb|Figure 3: Problem 3]]

== Connectedness ==
The following two DIY experiments are shown to better understand and visualize the physical consequences of electric fields.

'''Charged Rod and Aluminum Can'''

In our first example we set up an experiment using two charged rods placed to the left and right of an aluminum can, distanced by a length d. If one of the rods is positively charged and the other is negatively charged, what will the can do? Because the positively charged rod induces a negative charge on the left side of the can, creating an attractive force between the rod and the can, and the negatively charged rod induces an equal positive charge on the right side of the can, which creates an attractive force between the can and that rod, the net force on the can is zero. Thus the can will stay still. The setup is depicted in the image below.

[[File:plusq.png|300px|center|thumb|Figure 4: apparatus diagram with +q]]

Next, lets consider a setup but with both rods having equal positive charge, as shown in the image below. What will the can do in this situation?

[[File:minusq.png|300px|center|thumb|Figure 5: apparatus diagram with -q]]

Again the can will also stay still, but this time it is because the polarization force between two objects is always attractive.

So in what scenario will the can move, and what time of movement will the can exhibit? Considering the first setup, imagine this time we initially touch the negatively charged rod and the can for a brief moment. Holding the rods at equal distance on either side of the can, the can will now roll toward the positively charged rod. This is because the can acquires a net negative charge after being touched, so it is then attracted to the positively charged rod.

'''Charged Rod and Pith Ball'''

Another DIY experiment that demonstrates the effects of an electric field is shown in the video embedded below, which depicts the interaction between an initially neutral pith ball hanging on a string from a stand, and a charged rod.

[http://www.youtube.com/watch?v=aeiqw81kGio Interaction between a Charged Rod and Pith Ball]

== History ==

Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."

== See also ==

The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "[[Electric Field]]" for more information.

=== Further Reading ===

The page on electric fields: [[Electric Field]]

=== External Links ===
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html

http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp

https://pages.uncc.edu/phys2102/online-lectures/chapter-02-electric-field/2-4-electric-field-of-charge-distributions/example-1-electric-field-of-a-charged-rod-along-its-axis/

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml

== References ==

https://www.glowscript.org/#/

https://rhettallain_gmail_com.trinket.io/intro-to-electric-and-magnetic-fields#/electric-fields/multiple-charges

https://www.youtube.com/watch?v=BBWd0zUe0mI

(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)

Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15

[[Category: Electric Field]]