Physics Book - User contributions [en]

Gauss's Law

2024-04-14T23:54:20Z

Adendukuri7:

'''Claimed by James Swope (Fall 2020), Edited by Nabiha Ahsan (Fall 2021), Edited by Tyler Morgan(Fall 2023), Edited by Anshu Dendukuri(Spring 2024)'''


Gauss's Law is a very powerful law that spans a diverse array of fields, with applications in physics, mathematics, chemistry, and engineering, among others. Along with [[James Maxwell]]'s other three equations, Gauss's Law forms the foundation of classical electrodynamics.

==The Main Idea==

[[File:Gaussian Box Ex.jpg|thumb|Three-dimensional Gaussian surface]]
Picture a three-dimensional solid object in space that does not have any charges around it. Suppose that you discover an electric field is directed radially outward from all faces of the object. You then measure the magnitude of the electric field at each face. Without being able to open the object and determine its charge distribution, how might you determine the charge inside the object?

This simple situation illustrates one basic application of Gauss's Law: Using the measured electric field flowing through the faces of a closed surface to determine the charge that lies inside.

===A Mathematical Model===

Gauss's Law elegantly relates the net charge enclosed within a ''Gaussian surface'' to the patterns of electric field that flow over its faces ([[Electric Flux]]). A Gaussian surface is any surface belonging to a closed three-dimensional object. 
In words, the law states that the net electric flux outside a surface is equal to the ratio of the total enclosed charge inside the surface to the permittivity of free space. Symbolically, this is written as the area integral

<math>\oint \vec E \cdot \hat{n} ~dA = \frac{\sum Q_{inside}}{\varepsilon_{0}}</math>

where <math> \vec E </math> is the net electric field acting through the surface, <math> \hat{n}</math> is the unit normal vector perpendicular to the face of the surface, and <math> dA </math> is the surface area of the face being examined.



From the definition of [[Electric Flux]], Gauss's Law may also be written equivalently as

<math> \Phi_{el} = \frac{Q_{enclosed}}{\varepsilon_{0}}</math>

====Relation to Coulomb's Law====

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

====Relationship Between Charge and Flux====

We know from the previous introduction of flux that there is a relationship between the electric flux on a closed surface and the charges inside the surface. However, we would like to figure out what that particular factor is.

We start by considering a point charge of <math>+Q</math> enclosed by an imaginary spherical shell.

[[File:gaussv2212.jpg]]

Everywhere on the imaginary shell, the electric field produced is parallel to the normal unit vector. Cos(0) = 1, so the dot product is simply the magnitude of the electric field. Recall that the surface area of the imaginary sphere is <math>4 \pi r^{2}</math>. With Coulomb's Law and the surface area of a sphere, we get that electric flux is equal to <math>\frac{+Q}{\varepsilon_{0}}</math>. This implies that the factor is <math>\frac{1}{\varepsilon_{0}}</math>.

Note that if the point charge was negative, the electric field would still be parallel, just opposite direction. Cos(180) = -1 so the dot product would be -1 times the magnitude of the electric field.

To understand this law more fully, one can look at the diagram below. As can be seen on the left side of this diagram, change in flux equals electric field multiplied by change in area. This is the basis for the equation ф=∮E da. When you integrate the change of A, you get the flux.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, in integral form, the formula for this Law is the electric flux equals the total charge contained by a closed surface, divided by the permittivity (<math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>).

[[File:Adc2dff3156800a39ef0a9df76a7d868.png]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and <math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>.

Below is a brief summary of what has been covered.

[[File:Gauss_14.jpg]]

==A Computational Model==

An extremely rudimentary explanation of Gauss's Law is: https://www.youtube.com/watch?v=f2Cccp6XBUY

To visualization of Gauss's law with computational model: https://www.youtube.com/watch?v=i5N36mWsdGo

Experimental demonstration of Gauss's law: https://www.youtube.com/watch?v=r3w4FAUOLuM

Visual explaination of Gauss Divergence Theorem(Khan Academy): https://www.youtube.com/watch?v=XyiQ2dwJHXE

To view the applications of Gauss's law in a coding setting (Python GLowScript): https://trinket.io/glowscript/f618920f61

==Examples==

===Easy Examples===
====Easy Example 1====

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|400px]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|400px]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss fl.jpg|400px]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|600px]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

====Easy Example 2====
Consider a point charge <math> Q = +2 \, \text{μC} </math> enclosed by a spherical Gaussian surface with a radius of <math> r = 10 \, \text{cm} </math>. Calculate the electric flux through the surface.

**Solution:**
Using Gauss's Law, the electric flux <math>( \Phi_{el} )</math> through a closed surface is given by <math> \Phi_{el} = \frac{Q_{enclosed}}{\varepsilon_{0}} </math>. For a point charge enclosed by a spherical surface, the electric flux is simply <math> \frac{Q}{\varepsilon_{0}} </math>. Substituting the given values:

<math> \Phi_{el} = \frac{2 \times 10^{-6} \, \text{C}}{8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2} </math>

===Intermediate Examples===

====Example 1====
The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg|600px]]

====Example 2====
The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg|600px]]

====Example 3====
In order to apply Gauss's Law, it is important to be certain you are working with a closed surface, then set electric flux equal to the internal field divided by the permittivity (<math>\varepsilon_{0} = 8.854187817...×10^{-12} ~F*m^{-1}</math>). An example of this Law being applied can be found below.

[[File:Gauss_law3.png|600px]]

====Example 4====
In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent. The equation Ф=∮B da=0 helps explain this when considering that Ф is the same in both equations.

[[File:Gauss's_Law1234.jpeg|600px]]

===Hard Examples===

====Example 1====

Inside a non-uniformly charged sphere with a volume charge density <math>\rho(r) = \rho_0 r^2 </math> for <math>0 \leq r \leq R </math>, where <math>\rho_0</math> is a constant and R is the radius of the sphere. Find the electric field at a distance r from the center, where r is between 0 and R, using Gauss's Law.

=====Solution:=====
To find the electric field inside the sphere, create a Gaussian surface and take advantage of the spherical symmetry. The charge enclosed within a Gaussian sphere of radius <math>r</math> is <math>\frac{4}{3}\pi \rho_0 r^3 </math>. Gauss's Law then gives <math>E = \frac{\rho_0 r}{3\varepsilon_0} </math>.

====Example 2====

A long, straight wire carries a non-uniform linear charge density <math> \lambda(z) = \lambda_0 e^{-az} </math>, where <math>\lambda_0</math> and a are constants, and z is the distance along the wire. Determine the electric field at a distance r from the wire using Gauss's Law.

=====Solution:=====
Set up a Gaussian cylindrical surface around the wire to take advantage of the cylindrical symmetry. The total charge enclosed within the Gaussian surface is found by integrating the linear charge density along the length of the cylinder. Gauss's Law then gives <math>E = \frac{\lambda_0}{2\pi\varepsilon_0 r} \int_{-\infty}^{\infty} e^{-az} \, dz </math>. Solving this integral yields the expression for the electric field as a function of r.

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, we deal with many statistical distributions. A very important fundamental distribution is the Gaussian distribution or the Normal Distribution. With inference, the Gaussian Distribution comes up in confidence intervals for single statistics. With comparison inference, like finding the pairwise difference between statistics, generally we use other distributions such as the T-Distribution. However, the T-Distribution approximates the Gaussian distribution with degrees of freedom greater than 29.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

Additional to many engineering majors. Gauss's Law has applications in many different STEM job fields. Engineering majors learn about the different Gaussian laws in college and apply them in there jobs. Millions of people are applying his laws to the world and will continue to do so. Gauss has had a huge impact on the world and will continue to have a big impact on the future. Gauss has done great work with his work.

==History==

[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg||Portrait of Gauss]]

Gauss's Law was first 'discovered' in 1773, by the great mathematician Joseph-Louis Lagrange. It was further developed by Carl Friedrich Gauss in 1813.

Gauss was a prolific German mathematician and physicist. He made significant contributions to a variety of fields in mathematics and physics. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians".

== See also ==
===Further Reading===

Gauss's Law is tied in closely with the other of Maxwell's equations that can be found here in the Physics Book.

http://physicsbook.gatech.edu/Gauss%27s_Flux_Theorem

http://physicsbook.gatech.edu/Faraday%27s_Law

http://physicsbook.gatech.edu/Magnetic_Flux

http://physicsbook.gatech.edu/Ampere%27s_Law

===External links===

http://physics.info/law-gauss/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

https://en.wikipedia.org/wiki/Gauss%27s_law

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://physicscatalyst.com/elec/guass_0.php

==References==

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

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Gauss_Law.pdf

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

[[Category: Electric Field]]
[[Category: Fields]]
[[Category: Gauss's Law]]
[[Category: Maxwell's Equations]]

Understanding Fundamentals of Current, Voltage, and Resistance

2024-04-14T01:40:12Z

Adendukuri7:

'''Created and Edited by Kenneth (Alex) Jenkins - Fall 2023. Edited by Anshu Dendukuri - Spring 2024'''

The central concept in understanding the fundamentals of current, voltage, and resistance is unraveling the essential principles that govern the flow of electric charge. Current represents the rate of this flow, voltage signifies the driving force behind it, and resistance encapsulates the opposition encountered in the circuit. A grasp of these fundamentals is crucial for navigating the intricacies of electrical systems and technology.

==Ohm's Law==
[[Ohm's Law]], a fundamental principle in electrical engineering, establishes a foundational relationship between resistance, voltage, and current in a circuit. Named after the German physicist [[Georg Ohm]], the law states that the current passing through a conductor between two points is directly proportional to the voltage across the two points, given a constant temperature. Mathematically expressed as <math>{I = \frac{V}{R}}</math>, where I is the current in amperes, V is the voltage in volts, and R is the resistance in ohms, Ohm's Law is instrumental in unraveling the dynamic interplay between these three essential electrical parameters. This law provides a straightforward framework for understanding how changes in voltage or resistance influence the flow of current, and vice versa. Mastery of Ohm's Law is indispensable in analyzing and designing electrical circuits, serving as a cornerstone for engineers and enthusiasts as they navigate the intricate relationships among resistance, voltage, and current in the realm of electronics.

[[File:OhmsTriangle.jpg]]
<br>''Image Source: Kenneth Jenkins''

==Current==
Current, in the realm of electrical circuits, refers to the flow of electric charge through a conductor. It is the rate at which electrons move along a closed path, commonly a wire or circuitry. Measured in amperes (A), current is a fundamental concept in understanding the dynamic behavior of electricity. The flow of electrons is driven by the electric potential difference, or voltage, which propels them from areas of higher potential to lower potential. Visualizing current involves picturing the movement of these charged particles, akin to a river of electrons streaming through the conductive pathways of a circuit. Whether in the context of powering household appliances or enabling complex electronic devices, a clear comprehension of current is pivotal for navigating the principles that underpin the functioning of electrical systems.

==Voltage==
Voltage, within the realm of electrical systems, is a measure of electric potential difference between two points in a circuit. It represents the force that propels electric charges, typically electrons, to move through a conductor. Measured in volts (V), voltage serves as the driving factor behind the flow of current. It can be likened to the pressure in a water pipe that dictates the movement of water molecules; similarly, voltage dictates the movement of electric charges. Higher voltage implies a greater force pushing the charges, while lower voltage corresponds to a less forceful push. Understanding voltage is pivotal in comprehending the dynamics of electrical circuits, as it influences the rate and direction of the electric current, forming a foundational concept in the broader study of electrical engineering and technology.

==Resistance==
Resistance, in the realm of electrical systems, is the property that hinders the flow of electric current. It is a measure of the opposition encountered by the flow of electrons as they traverse through a conductor. This opposition leads to the conversion of electrical energy into heat. Resistance is quantified in ohms (Ω), and it is a critical factor in determining the behavior of circuits. Materials with high resistance impede the flow of current more strongly than those with low resistance. Resistors, specific components designed to introduce resistance intentionally, are commonly employed in circuits to regulate and control the flow of current, demonstrating the essential role that resistance plays in shaping the characteristics and functionality of electrical systems. A nuanced understanding of resistance is vital for engineers and enthusiasts alike as they design and optimize circuits for various applications.

==Analogy to Water==

Circuitry can be conceptually likened to the flow of water through a network of pipes, providing an insightful analogy that simplifies the complex dynamics of electrical systems. In this analogy, electrical circuits serve as the conduits for the flow of electrons, analogous to water molecules coursing through pipes. The fundamental principles governing the behavior of both water and electrical circuits draw intriguing parallels, offering a relatable framework for understanding the intricate world of electronics.

Just as water moves from a source to various destinations through a network of interconnected pipes, electrical circuits facilitate the flow of electric current from a power source to multiple components within a system. The pipes themselves can be equated to conductive materials, such as copper wires, that guide the electrons along a predetermined path. Much like the pressure applied to water influencing its movement through pipes, voltage serves as the driving force behind the flow of electrons in a circuit.

Resistors within an electrical circuit find an analogy in the narrowing of pipes or the introduction of obstacles that impede the smooth passage of water. These resistive elements in a circuit limit the flow of electric current, generating heat in a manner akin to the friction-induced warmth observed in constricted water pipes. Capacitors and inductors, on the other hand, can be compared to the storage tanks and coiled sections in a water system, respectively. Capacitors store electrical energy, analogous to water reservoirs, while inductors store energy in a magnetic field, echoing the potential energy stored in coiled pipes.

The analogy of circuits as conduits for the flow of electrons, similar to water coursing through pipes, serves as a didactic tool, allowing individuals to grasp the intricacies of electrical systems through a familiar and tangible metaphor. Just as plumbing systems distribute water efficiently, electrical circuits enable the controlled movement of electrons, powering a myriad of devices and technologies that have become integral to our modern way of life.

===Examples of the Water/Pipe Analogy===

For a demonstration, please consider watching this video produced in collaboration with the EATON company.

<youtube>Z_s3TmYQAuc</youtube>

The electronics retailer Sparkfun also made a video in similar format to EATON's, covering the topics in a much more in depth way.

<youtube>8jB6hDUqN0Y</youtube>

==Connectedness to Applications Outside of Physics==
The concepts encapsulated in Ohm's Law transcend the confines of physics, resonating profoundly in the realm of engineering and various practical applications. Engineering disciplines, particularly electrical and electronic engineering, heavily rely on the principles outlined in Ohm's Law to design, analyze, and optimize a myriad of systems and devices. Understanding the interplay between resistance, voltage, and current allows engineers to predict and control the behavior of electrical circuits, ensuring the efficient and safe operation of electronic components.

In the field of electrical engineering, Ohm's Law is a cornerstone for designing circuits with specific performance characteristics. Engineers leverage the law to determine the appropriate resistances needed for components, calculate voltage drops across various elements, and establish the current requirements for optimal functionality. Whether designing intricate integrated circuits or power distribution systems, the principles of Ohm's Law provide a fundamental framework for engineers to achieve desired electrical outcomes.

Beyond traditional engineering disciplines, Ohm's Law finds application in diverse technological domains. For instance, in the burgeoning field of renewable energy, such as solar power systems, understanding the relationship between voltage, current, and resistance is crucial for designing efficient energy conversion and storage systems. Similarly, in telecommunications, where signal integrity is paramount, the principles of Ohm's Law guide the design of communication networks, ensuring reliable transmission of information through cables and electronic components.

In essence, the universality of Ohm's Law extends its influence into a spectrum of engineering applications, shaping the way professionals approach challenges in fields ranging from electronics and telecommunications to renewable energy. Its principles serve as a practical and indispensable tool, providing a systematic approach to understanding and manipulating the behavior of electrical systems in the pursuit of technological innovation.

==History==
In the early 19th century, the study of electricity was in its infancy, and it was during this time that German physicist [https://www.physicsbook.gatech.edu/Georg_Ohm Georg Simon Ohm] made groundbreaking contributions, laying the foundation for what would become Ohm's Law. Ohm, born in Erlangen, Bavaria, in 1789, embarked on his scientific journey in an era marked by fervent exploration into the nature of electricity.

Georg Simon Ohm's pioneering work culminated in 1827 when he published his seminal treatise "Die galvanische Kette, mathematisch bearbeitet" ("The Galvanic Circuit Investigated Mathematically"). In this work, Ohm unveiled the relationship between voltage, current, and resistance, providing a mathematical formula that encapsulated the fundamental principles. His revolutionary concept, known today as Ohm's Law, asserted that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance it offers.

Ohm's Law addressed a pressing need in the scientific community at the time, offering a quantitative framework to comprehend and manipulate electrical circuits. It was a watershed moment that transformed electricity from a mysterious force into a quantifiable and predictable phenomenon.

The practical implications of Ohm's Law began to unfold as the fields of physics and engineering evolved. Michael Faraday's groundbreaking work in electromagnetic induction in the early 19th century and James Clerk Maxwell's formulation of Maxwell's Equations in the mid-19th century further enriched the understanding of electricity, providing context to Ohm's Law.

As electrical science progressed, so did the understanding of voltage, current, and resistance. The introduction of the telegraph in the mid-19th century and the subsequent development of electrical power distribution systems in the late 19th century underscored the practical utility of these concepts. Engineers and scientists across the globe, including luminaries like Thomas Edison and Nikola Tesla, applied the principles elucidated by Ohm to propel the electrical revolution, shaping the modern technological landscape.

In summary, Ohm's Law emerged as a cornerstone in the historical tapestry of electrical science, catalyzing a transformative shift in how electricity was understood and harnessed. Its profound impact resonates through centuries, influencing the trajectory of technological progress and laying the groundwork for the sophisticated electrical systems that define our contemporary world.

== See also ==
<youtube>mc979OhitAg&list=PLWv9VM947MKi_7yJ0_FCfzTBXpQU-Qd3K‎</youtube>

===Further reading===
Access to IEEE Xplore (a digital library covering all aspects of Electrical and Computer Engineering) is provided for free through the Georgia Tech Library. The library offers various free ebooks covering topics of [https://ieeexplore.ieee.org/book/5521805 Power Systems], and [https://ieeexplore.ieee.org/book/9549029 Electricity and Electronics Fundamentals].

==References==
#[https://www.fluke.com/en-us/learn/blog/electrical/what-is-ohms-law What is Ohm’s Law? (n.d.). Fluke LLC.]

[[Category:Circuits]]