Physics Book - User contributions [en]

Combining Electric and Magnetic Forces

2018-11-25T06:13:22Z

Lpimentel3: /* Connectedness */

Claimed by Luis Pimentel Fall 2018
Edited the Velocity Selector section. Went into more detail about why this works and created a VPython trinket demonstrating and visualizing a velocity selector for particles.

Though the pattern in which electric and magnetic forces interact with particles is observably different, their effects can be quantitatively be compared. The principle of adding the two functions of force as a net force is one that now serves as a fundamental principle of electromagnetics. It serves as a building block for many important Laws such as [[Hall Effect]] and [[Motional Emf]].

==The Main Idea==

If a charged particle within an electric field is moving in a magnetic field, the particle is subject to an [[Electric Force]] and a [[Magnetic Force]]. The net force on the particle is called the [[Lorentz Force]], which is the sum of electric and magnetic forces.

===A Mathematical Model===

====Electric Forces====

[[File:ElectricForces.jpg|thumb| '''Figure 1.''' An electric force acts in a pattern parallel to the electric field, pointing radially inward or outward of a particle. The direction depends on the signs of the interacting charged particles. ]]

• A particle being acted upon by an electric force will move in a straight line, in the path, or negative path depending on charge, of the the electric field line (See '''Figure 1''') .

• Electric fields point in a direction radially outward/ inward of a charged particle. There are four possible scenarios for the interaction of 2 charged particles:

:1. A negatively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially outward from p2
:2. A positively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially inward toward p2
:3. A negatively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially inward toward p1
:4. A positively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially outward from p1

• The electric force formula: <math>\vec {F}_{E}=q\vec E </math>
:- Force on the observed particle is determined by the interaction of the charge of the observed particle and the electric field created by other charged particles.

====Magnetic Forces====

[[File:Magnetic Force Lines.jpg|thumb| '''Figure 2.''' Magnetic Fields follow a helical pattern ]]
[[File:RightHandRule.jpg|thumb| '''Figure 3.''' Magnetic Force Right Hand Rule]]

• The magnetic force on a charged particle is orthogonal to the magnetic field.

• The particle must be moving with some velocity for a magnetic force to be present.

• Particles move perpendicular to the magnetic field lines in a helical manner (See '''Figure 2''')

• To find the magnetic force, you can use the Right Hand Rule as follows (See '''Figure 3'''):

:1. Thumb in direction of the velocity
:2. Fingers in the direction of the magnetic field
:3. Your palm will face in the direction of the Magnetic Force

• The magnetic force formula: <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>
:- q is the charge of the moving charge, including its sign
:- <math>\vec v</math> is the velocity of the moving charge
:- <math>\vec B</math> is the applied magnetic field, in Tesla
:- Note: if <math>\vec v</math> and <math>\vec B</math> are parallel to each other, <math> {\vec {F}_{M} = 0} </math> (<math> {\vec {A}\times\vec {B} = |\vec A||\vec B|sin(θ) = 0} </math>)

====Electric and Magnetic Forces Combined====
[[File:Velocity selector.gif|thumb| '''Figure 4.''' The electric field, magnetic field, and velocity vector are all perpendicular to each other ]]

• The Lorentz Force formula:
:<math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{M}} </math>
:<math> {\vec {F}_{net} = q\vec E + q\vec {v}\times\vec {B}} </math>

• When the net force is equal to zero, the velocity stays constant.
:<math> {\vec {F}_{E} = \vec {F}_{M}} </math>
:<math> {q\vec E = q\vec {v}\times\vec {B}} </math>

As seen in '''Figure 4''' , when the net forces acting on a particle are balanced the electric field, magnetic field, and velocity vector are all perpendicular to each other. The electric and magnetic forces are equal but opposite. When forces are not balanced the trajectory of the the particle will change.

The Lorentz Force calculation is now a fundamental principle of electromagnetism.

===A Computational Model===

Following are diagrams which display a uniform electric field in the +x direction and a uniform magnetic field in +y direction for a proton and an electron, with varying velocities.

The force equations and the right hand rule can both be used to determine the directions of the forces:
:- According to <math> {\vec {F}_{E} = q\vec E} </math>, <math> {\vec {F}_{Eproton}} </math> points in the direction of <math> \vec E </math> and <math> {\vec {F}_{Eelectron}} </math> points in opposite direction of <math> \vec E </math>.
:- According to <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>, <math> \vec {F}_{M} = 0 </math> when <math> \vec v </math> is parallel to <math> \vec B </math>. The direction of <math> \vec {F}_{M} </math> can be determined by the cross multiplication of <math> \vec v </math> and <math> \vec B </math> and by the sign of <math> q </math>.

• '''Proton at rest:'''
:[[File:protonr2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +x direction:'''
:[[File:protonx2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: +z

• '''Proton moving in +y direction:'''
:[[File:protony.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +z direction:'''
:[[File:protonz.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: -x

• '''Electron at rest:'''
:[[File:electronr2.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +x direction:'''
:[[File:electronx.png]]
:- Direction of electric force:-x
:- Direction of magnetic force: -z

• '''Electron moving in +y direction:'''
:[[File:electrony.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +z direction:'''
:[[File:electronz.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: +x

==Examples==

===Simple===

A proton is moving with velocity 7e8 in the +x direction. The trajectory of the proton is constant. There is an electric field in the area of 3.6e7 in the +y direction. Calculate the direction and magnitude of the magnetic field acting on the particle?

:''Solution:''
:Step 1: <math> {|q\vec E| = |q\vec v\vec B|} </math>
:Step 2: <math> {\vec {E} = \vec {v}\vec {B}} </math>
:Step 3: <math> {\vec {B} = \frac {\vec {E}} {\vec {v}} = \frac {3.6e7} {7e8}} </math>

:Answer: <math> {\vec {B} = 0.051 T} </math>

The magnetic field is in the +z direction.

===Middling===

At a particular instant, a proton is moving with velocity <0,5e5,0> m/s and an electron is moving with velocity <-4.2e2,0,0> m/s. The electron is located 1.4e-3 m below the proton (in the -y direction). Determine the net force on the electron due to the proton.

:''Solution:''
:Step 1: <math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{B}} </math>
:Step 2: <math> {\vec {F}_{net} = \vec {F}_{E} + 0 = q\vec {E}} </math> (At the electron's location, <math> \vec B = 0 </math> because the velocity of the proton is parallel to <math> \hat{r} </math>)
:Step 3: <math> {\vec {E} = \frac {1} {4πεo} \frac {q} {r^2} \hat{r}} </math> (<math> \hat{r} = <0,-1,0> and r = 1.4e-3 m</math>)
:Step 4: <math> \vec E = <0,-7.35e-4,0> N/C </math>
:Step 5: <math> {\vec {F}_{net} = -e \vec E = <0,1.18e-22,0> N} </math>

===Difficult===
[[File:Middle_Example_A.jpg|thumb| '''Figure 5.''' Middle example diagram. ]]

A copper bar of length L and zero resistance slides at a constant velocity, v. There is a uniform magnetic field, B, directed into the page. A voltmeter is connected across a resistor, R, and reads ΔV. See '''Figure 5'''. Determine the direction of the magnetic force on the diagram and the current through the resistor. Your answer should be in terms of the given variables.

[[File:Middle_Example_B.jpg|thumb| '''Figure 6.''' Middle example solution. ]]

:''Solution:''
:Step 1: Direction of the Magnetic Force - Use right hand rule or <math> {\vec v \times\vec B} </math>. See '''Figure 6'''.
:Step 2: Current through the resistor -
::: <math> {ΔV}_{round trip} = 0 </math>
::: <math> {{ΔV}_{round trip} + motional emf - {ΔV}_{resistor} = 0} </math>
::: <math> motional emf = IR </math>
::: <math> I = \frac {motional emf} {R} </math>
::: <math> {|q\vec E| = |q\vec v\vec B|} </math>, <math> {|ΔV| = IR} </math>
::: <math> {qE = qvB} </math>
::: <math> E = vB </math>
::: <math> motional emf = |ΔV| = E*ΔL = vBL </math>
::: <math> {I = \frac {vBL} {R}} </math>

==Connectedness==

The Lorentz Force principle has been a component in many modern day inventions and critical building block for many physics principles. With known forces, we can predict the very important figure, the velocity and trajectory of a moving particle.

===Applications===

'''''Velocity Selector'''''
[[File:Velocityselector2.png|thumb| '''Figure 7.''' Illustration of a Velocity Selector ]]

The Velocity Selector is a device used to filter particles based on their velocity. A Velocity Selector uses controlled, perpendicular, electric and magnetic fields to filter certain charged particles (See '''Figure 7''' ). These electric and magnetic fields exert a Lorentz force on the particle. For the particle to remain unaffected, the Lorentz force must be zero. Therefore, we have:

<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B} = 0 </math>

<math> \lvert q\vec{E}\rvert = \lvert q\vec{v} ⨯ \vec{B}\rvert </math>

<math> qE= qvB </math>

<math> v = E/B </math>

From this relationship we can adjust to our electric and magnetic fields to pass particles with a desired speed of a narrow band through a target area. Particles at a desired speed will remain unaffected while other particles with undesired speeds will be deflected. This technique is used in technologies such as electron microscopes and spectrometers.

The following VPython trinket module is a demonstration of a velocity selector. Two particles of different speeds pass through an area with perpendicular magnetic and electric fields. Within this trinket, you can change the initial values of the velocity selector and particle velocities to see how this affects the particles' motion as they try to pass through a circular area.

[[File:velocitySelector3Stages.jpg|thumb| '''Figure 7.1.''' SImulation of a Velocity Selector ]]

[https://trinket.io/glowscript/d3067081b7 Velocity Selector Trinket]

'''''Electric Motor'''''

An electric motor is a device that uses the Lorentz force to convert electric energy into mechanical energy. Using the [[magnetic torque]] principle, electric energy is created by using the magnetic field of a magnet. The [[torque]] laws are based off the principles of the net electric and magnetic forces.

===Other related topics===

Here are other principles that use the net force of magnetic and electric forces as a building block:

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

==History==

The relationship between electric and magnetic forces was first questioned in the mid-18th century when Johann Tobias Mayer (1760) and Henry Cavendish (1762) proposed that the force on magnetic poles and electrically charged objects followed the inverse-square law, which was proven to be true by Charles-Augustin in 1784.

Following Michael Faraday's proposal of the concept of electric and magnetic fields, James Clerk Maxwell was first to mathematically prove the concepts. In 1865, Maxwell's field equations consisted of some form of the Lorentz force equation, but at the time it was not clear how it related to forces on charged moving particles. J.J Thomson was the first to attempt a derivation of Maxwell's field equations, and he derived a basic form of the formula for the electromagnetic forces on a charged moving particle in relation to the properties of the particle and its external fields. In 1892, Hendrik Lorentz corrected mistakes of the old formula and derived the modern form of the equation, which contains the forces due to both electric and magnetic fields.

== See also ==

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

===Further reading===

Books, Articles or other print media on this topic

[http://web.mit.edu/sahughes/www/8.022/lec10.pdf | MIT Physics notes on Lorentz force]

===External links===
Great youtube videos on Lorentz Force Law:
[https://www.youtube.com/watch?v=gINzRCOOs-8 |Lorentz Force Law Video 1]
[https://www.youtube.com/watch?v=PK6sEF9SNjw | Lorentz Force Law Video 2]

==References==

Boundless. “Electric vs. Magnetic Forces.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 05 Dec. 2015 from https://www.boundless.com/physics/textbooks/boundless-physics-textbook/magnetism-21/motion-of-a-charged-particle-in-a-magnetic-field-158/electric-vs-magnetic-forces-554-11176/

Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Vol. 2. Hoboken, NJ: Wiley, 2015. 812-814. Print.

All images found on google image search:
https://en.wikipedia.org/wiki/Magnetic_field
https://en.wikipedia.org/wiki/Wien_filter
http://aplusphysics.com/wordpress/regents/em/electric-field/

[[Fields]]

File:VelocitySelector3Stages.jpg

2018-11-25T06:11:08Z

Lpimentel3:

Combining Electric and Magnetic Forces

2018-11-25T06:10:25Z

Lpimentel3: /* Applications */

Claimed by Luis Pimentel Fall 2018
Edited the Velocity Selector section. Went into more detail about why this works and created a VPython trinket demonstrating and visualizing a velocity selector for particles.

Though the pattern in which electric and magnetic forces interact with particles is observably different, their effects can be quantitatively be compared. The principle of adding the two functions of force as a net force is one that now serves as a fundamental principle of electromagnetics. It serves as a building block for many important Laws such as [[Hall Effect]] and [[Motional Emf]].

==The Main Idea==

If a charged particle within an electric field is moving in a magnetic field, the particle is subject to an [[Electric Force]] and a [[Magnetic Force]]. The net force on the particle is called the [[Lorentz Force]], which is the sum of electric and magnetic forces.

===A Mathematical Model===

====Electric Forces====

[[File:ElectricForces.jpg|thumb| '''Figure 1.''' An electric force acts in a pattern parallel to the electric field, pointing radially inward or outward of a particle. The direction depends on the signs of the interacting charged particles. ]]

• A particle being acted upon by an electric force will move in a straight line, in the path, or negative path depending on charge, of the the electric field line (See '''Figure 1''') .

• Electric fields point in a direction radially outward/ inward of a charged particle. There are four possible scenarios for the interaction of 2 charged particles:

:1. A negatively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially outward from p2
:2. A positively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially inward toward p2
:3. A negatively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially inward toward p1
:4. A positively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially outward from p1

• The electric force formula: <math>\vec {F}_{E}=q\vec E </math>
:- Force on the observed particle is determined by the interaction of the charge of the observed particle and the electric field created by other charged particles.

====Magnetic Forces====

[[File:Magnetic Force Lines.jpg|thumb| '''Figure 2.''' Magnetic Fields follow a helical pattern ]]
[[File:RightHandRule.jpg|thumb| '''Figure 3.''' Magnetic Force Right Hand Rule]]

• The magnetic force on a charged particle is orthogonal to the magnetic field.

• The particle must be moving with some velocity for a magnetic force to be present.

• Particles move perpendicular to the magnetic field lines in a helical manner (See '''Figure 2''')

• To find the magnetic force, you can use the Right Hand Rule as follows (See '''Figure 3'''):

:1. Thumb in direction of the velocity
:2. Fingers in the direction of the magnetic field
:3. Your palm will face in the direction of the Magnetic Force

• The magnetic force formula: <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>
:- q is the charge of the moving charge, including its sign
:- <math>\vec v</math> is the velocity of the moving charge
:- <math>\vec B</math> is the applied magnetic field, in Tesla
:- Note: if <math>\vec v</math> and <math>\vec B</math> are parallel to each other, <math> {\vec {F}_{M} = 0} </math> (<math> {\vec {A}\times\vec {B} = |\vec A||\vec B|sin(θ) = 0} </math>)

====Electric and Magnetic Forces Combined====
[[File:Velocity selector.gif|thumb| '''Figure 4.''' The electric field, magnetic field, and velocity vector are all perpendicular to each other ]]

• The Lorentz Force formula:
:<math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{M}} </math>
:<math> {\vec {F}_{net} = q\vec E + q\vec {v}\times\vec {B}} </math>

• When the net force is equal to zero, the velocity stays constant.
:<math> {\vec {F}_{E} = \vec {F}_{M}} </math>
:<math> {q\vec E = q\vec {v}\times\vec {B}} </math>

As seen in '''Figure 4''' , when the net forces acting on a particle are balanced the electric field, magnetic field, and velocity vector are all perpendicular to each other. The electric and magnetic forces are equal but opposite. When forces are not balanced the trajectory of the the particle will change.

The Lorentz Force calculation is now a fundamental principle of electromagnetism.

===A Computational Model===

Following are diagrams which display a uniform electric field in the +x direction and a uniform magnetic field in +y direction for a proton and an electron, with varying velocities.

The force equations and the right hand rule can both be used to determine the directions of the forces:
:- According to <math> {\vec {F}_{E} = q\vec E} </math>, <math> {\vec {F}_{Eproton}} </math> points in the direction of <math> \vec E </math> and <math> {\vec {F}_{Eelectron}} </math> points in opposite direction of <math> \vec E </math>.
:- According to <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>, <math> \vec {F}_{M} = 0 </math> when <math> \vec v </math> is parallel to <math> \vec B </math>. The direction of <math> \vec {F}_{M} </math> can be determined by the cross multiplication of <math> \vec v </math> and <math> \vec B </math> and by the sign of <math> q </math>.

• '''Proton at rest:'''
:[[File:protonr2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +x direction:'''
:[[File:protonx2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: +z

• '''Proton moving in +y direction:'''
:[[File:protony.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +z direction:'''
:[[File:protonz.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: -x

• '''Electron at rest:'''
:[[File:electronr2.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +x direction:'''
:[[File:electronx.png]]
:- Direction of electric force:-x
:- Direction of magnetic force: -z

• '''Electron moving in +y direction:'''
:[[File:electrony.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +z direction:'''
:[[File:electronz.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: +x

==Examples==

===Simple===

A proton is moving with velocity 7e8 in the +x direction. The trajectory of the proton is constant. There is an electric field in the area of 3.6e7 in the +y direction. Calculate the direction and magnitude of the magnetic field acting on the particle?

:''Solution:''
:Step 1: <math> {|q\vec E| = |q\vec v\vec B|} </math>
:Step 2: <math> {\vec {E} = \vec {v}\vec {B}} </math>
:Step 3: <math> {\vec {B} = \frac {\vec {E}} {\vec {v}} = \frac {3.6e7} {7e8}} </math>

:Answer: <math> {\vec {B} = 0.051 T} </math>

The magnetic field is in the +z direction.

===Middling===

At a particular instant, a proton is moving with velocity <0,5e5,0> m/s and an electron is moving with velocity <-4.2e2,0,0> m/s. The electron is located 1.4e-3 m below the proton (in the -y direction). Determine the net force on the electron due to the proton.

:''Solution:''
:Step 1: <math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{B}} </math>
:Step 2: <math> {\vec {F}_{net} = \vec {F}_{E} + 0 = q\vec {E}} </math> (At the electron's location, <math> \vec B = 0 </math> because the velocity of the proton is parallel to <math> \hat{r} </math>)
:Step 3: <math> {\vec {E} = \frac {1} {4πεo} \frac {q} {r^2} \hat{r}} </math> (<math> \hat{r} = <0,-1,0> and r = 1.4e-3 m</math>)
:Step 4: <math> \vec E = <0,-7.35e-4,0> N/C </math>
:Step 5: <math> {\vec {F}_{net} = -e \vec E = <0,1.18e-22,0> N} </math>

===Difficult===
[[File:Middle_Example_A.jpg|thumb| '''Figure 5.''' Middle example diagram. ]]

A copper bar of length L and zero resistance slides at a constant velocity, v. There is a uniform magnetic field, B, directed into the page. A voltmeter is connected across a resistor, R, and reads ΔV. See '''Figure 5'''. Determine the direction of the magnetic force on the diagram and the current through the resistor. Your answer should be in terms of the given variables.

[[File:Middle_Example_B.jpg|thumb| '''Figure 6.''' Middle example solution. ]]

:''Solution:''
:Step 1: Direction of the Magnetic Force - Use right hand rule or <math> {\vec v \times\vec B} </math>. See '''Figure 6'''.
:Step 2: Current through the resistor -
::: <math> {ΔV}_{round trip} = 0 </math>
::: <math> {{ΔV}_{round trip} + motional emf - {ΔV}_{resistor} = 0} </math>
::: <math> motional emf = IR </math>
::: <math> I = \frac {motional emf} {R} </math>
::: <math> {|q\vec E| = |q\vec v\vec B|} </math>, <math> {|ΔV| = IR} </math>
::: <math> {qE = qvB} </math>
::: <math> E = vB </math>
::: <math> motional emf = |ΔV| = E*ΔL = vBL </math>
::: <math> {I = \frac {vBL} {R}} </math>

==Connectedness==

The Lorentz Force principle has been a component in many modern day inventions and critical building block for many physics principles. With known forces, we can predict the very important figure, the velocity and trajectory of a moving particle.

===Applications===

'''''Velocity Selector'''''
[[File:Velocityselector2.png|thumb| '''Figure 7.''' Illustration of a Velocity Selector ]]

The Velocity Selector is a device used to filter particles based on their velocity. A Velocity Selector uses controlled, perpendicular, electric and magnetic fields to filter certain charged particles (See '''Figure 7''' ). These electric and magnetic fields exert a Lorentz force on the particle. For the particle to remain unaffected, the Lorentz force must be zero. Therefore, we have:

<math> \vec{F}_{Lorentz} = q\vec{E} + q\vec{v} ⨯ \vec{B} = 0 </math>

<math> \lvert q\vec{E}\rvert = \lvert q\vec{v} ⨯ \vec{B}\rvert </math>

<math> qE= qvB </math>

<math> v = E/B </math>

From this relationship we can adjust to our electric and magnetic fields to pass particles with a desired speed of a narrow band through a target area. Particles at a desired speed will remain unaffected while other particles with undesired speeds will be deflected. This technique is used in technologies such as electron microscopes and spectrometers.

The following VPython trinket module is a demonstration of a velocity selector. Two particles of different speeds pass through an area with perpendicular magnetic and electric fields. Within this trinket, you can change the initial values of the velocity selector and particle velocities to see how this affects the particles' motion as they try to pass through a circular area.

[[File:velocitySelector3Stages.jpg]]

[https://trinket.io/glowscript/d3067081b7 Velocity Selector Trinket]

'''''Electric Motor'''''

An electric motor is a device that uses the Lorentz force to convert electric energy into mechanical energy. Using the [[magnetic torque]] principle, electric energy is created by using the magnetic field of a magnet. The [[torque]] laws are based off the principles of the net electric and magnetic forces.

===Other related topics===

Here are other principles that use the net force of magnetic and electric forces as a building block:

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

==History==

The relationship between electric and magnetic forces was first questioned in the mid-18th century when Johann Tobias Mayer (1760) and Henry Cavendish (1762) proposed that the force on magnetic poles and electrically charged objects followed the inverse-square law, which was proven to be true by Charles-Augustin in 1784.

Following Michael Faraday's proposal of the concept of electric and magnetic fields, James Clerk Maxwell was first to mathematically prove the concepts. In 1865, Maxwell's field equations consisted of some form of the Lorentz force equation, but at the time it was not clear how it related to forces on charged moving particles. J.J Thomson was the first to attempt a derivation of Maxwell's field equations, and he derived a basic form of the formula for the electromagnetic forces on a charged moving particle in relation to the properties of the particle and its external fields. In 1892, Hendrik Lorentz corrected mistakes of the old formula and derived the modern form of the equation, which contains the forces due to both electric and magnetic fields.

== See also ==

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

===Further reading===

Books, Articles or other print media on this topic

[http://web.mit.edu/sahughes/www/8.022/lec10.pdf | MIT Physics notes on Lorentz force]

===External links===
Great youtube videos on Lorentz Force Law:
[https://www.youtube.com/watch?v=gINzRCOOs-8 |Lorentz Force Law Video 1]
[https://www.youtube.com/watch?v=PK6sEF9SNjw | Lorentz Force Law Video 2]

==References==

Boundless. “Electric vs. Magnetic Forces.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 05 Dec. 2015 from https://www.boundless.com/physics/textbooks/boundless-physics-textbook/magnetism-21/motion-of-a-charged-particle-in-a-magnetic-field-158/electric-vs-magnetic-forces-554-11176/

Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Vol. 2. Hoboken, NJ: Wiley, 2015. 812-814. Print.

All images found on google image search:
https://en.wikipedia.org/wiki/Magnetic_field
https://en.wikipedia.org/wiki/Wien_filter
http://aplusphysics.com/wordpress/regents/em/electric-field/

[[Fields]]

Combining Electric and Magnetic Forces

2018-11-25T05:33:10Z

Lpimentel3:

Claimed by Luis Pimentel Fall 2018
Edited the Velocity Selector section. Went into more detail about why this works and created a VPython trinket demonstrating and visualizing a velocity selector for particles.

Though the pattern in which electric and magnetic forces interact with particles is observably different, their effects can be quantitatively be compared. The principle of adding the two functions of force as a net force is one that now serves as a fundamental principle of electromagnetics. It serves as a building block for many important Laws such as [[Hall Effect]] and [[Motional Emf]].

==The Main Idea==

If a charged particle within an electric field is moving in a magnetic field, the particle is subject to an [[Electric Force]] and a [[Magnetic Force]]. The net force on the particle is called the [[Lorentz Force]], which is the sum of electric and magnetic forces.

===A Mathematical Model===

====Electric Forces====

[[File:ElectricForces.jpg|thumb| '''Figure 1.''' An electric force acts in a pattern parallel to the electric field, pointing radially inward or outward of a particle. The direction depends on the signs of the interacting charged particles. ]]

• A particle being acted upon by an electric force will move in a straight line, in the path, or negative path depending on charge, of the the electric field line (See '''Figure 1''') .

• Electric fields point in a direction radially outward/ inward of a charged particle. There are four possible scenarios for the interaction of 2 charged particles:

:1. A negatively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially outward from p2
:2. A positively charged particle (p1) is acting on a negatively charged particle (p2)
::- p2 feels force pointing radially inward toward p2
:3. A negatively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially inward toward p1
:4. A positively charged particle (p1) is acting on a positively charged particle (p2)
::- p2 feels force pointing radially outward from p1

• The electric force formula: <math>\vec {F}_{E}=q\vec E </math>
:- Force on the observed particle is determined by the interaction of the charge of the observed particle and the electric field created by other charged particles.

====Magnetic Forces====

[[File:Magnetic Force Lines.jpg|thumb| '''Figure 2.''' Magnetic Fields follow a helical pattern ]]
[[File:RightHandRule.jpg|thumb| '''Figure 3.''' Magnetic Force Right Hand Rule]]

• The magnetic force on a charged particle is orthogonal to the magnetic field.

• The particle must be moving with some velocity for a magnetic force to be present.

• Particles move perpendicular to the magnetic field lines in a helical manner (See '''Figure 2''')

• To find the magnetic force, you can use the Right Hand Rule as follows (See '''Figure 3'''):

:1. Thumb in direction of the velocity
:2. Fingers in the direction of the magnetic field
:3. Your palm will face in the direction of the Magnetic Force

• The magnetic force formula: <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>
:- q is the charge of the moving charge, including its sign
:- <math>\vec v</math> is the velocity of the moving charge
:- <math>\vec B</math> is the applied magnetic field, in Tesla
:- Note: if <math>\vec v</math> and <math>\vec B</math> are parallel to each other, <math> {\vec {F}_{M} = 0} </math> (<math> {\vec {A}\times\vec {B} = |\vec A||\vec B|sin(θ) = 0} </math>)

====Electric and Magnetic Forces Combined====
[[File:Velocity selector.gif|thumb| '''Figure 4.''' The electric field, magnetic field, and velocity vector are all perpendicular to each other ]]

• The Lorentz Force formula:
:<math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{M}} </math>
:<math> {\vec {F}_{net} = q\vec E + q\vec {v}\times\vec {B}} </math>

• When the net force is equal to zero, the velocity stays constant.
:<math> {\vec {F}_{E} = \vec {F}_{M}} </math>
:<math> {q\vec E = q\vec {v}\times\vec {B}} </math>

As seen in '''Figure 4''' , when the net forces acting on a particle are balanced the electric field, magnetic field, and velocity vector are all perpendicular to each other. The electric and magnetic forces are equal but opposite. When forces are not balanced the trajectory of the the particle will change.

The Lorentz Force calculation is now a fundamental principle of electromagnetism.

===A Computational Model===

Following are diagrams which display a uniform electric field in the +x direction and a uniform magnetic field in +y direction for a proton and an electron, with varying velocities.

The force equations and the right hand rule can both be used to determine the directions of the forces:
:- According to <math> {\vec {F}_{E} = q\vec E} </math>, <math> {\vec {F}_{Eproton}} </math> points in the direction of <math> \vec E </math> and <math> {\vec {F}_{Eelectron}} </math> points in opposite direction of <math> \vec E </math>.
:- According to <math> {\vec {F}_{M} = q\vec {v}\times\vec {B}} </math>, <math> \vec {F}_{M} = 0 </math> when <math> \vec v </math> is parallel to <math> \vec B </math>. The direction of <math> \vec {F}_{M} </math> can be determined by the cross multiplication of <math> \vec v </math> and <math> \vec B </math> and by the sign of <math> q </math>.

• '''Proton at rest:'''
:[[File:protonr2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +x direction:'''
:[[File:protonx2.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: +z

• '''Proton moving in +y direction:'''
:[[File:protony.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: no magnetic force

• '''Proton moving in +z direction:'''
:[[File:protonz.png]]
:- Direction of electric force: +x
:- Direction of magnetic force: -x

• '''Electron at rest:'''
:[[File:electronr2.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +x direction:'''
:[[File:electronx.png]]
:- Direction of electric force:-x
:- Direction of magnetic force: -z

• '''Electron moving in +y direction:'''
:[[File:electrony.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: no magnetic force

• '''Electron moving in +z direction:'''
:[[File:electronz.png]]
:- Direction of electric force: -x
:- Direction of magnetic force: +x

==Examples==

===Simple===

A proton is moving with velocity 7e8 in the +x direction. The trajectory of the proton is constant. There is an electric field in the area of 3.6e7 in the +y direction. Calculate the direction and magnitude of the magnetic field acting on the particle?

:''Solution:''
:Step 1: <math> {|q\vec E| = |q\vec v\vec B|} </math>
:Step 2: <math> {\vec {E} = \vec {v}\vec {B}} </math>
:Step 3: <math> {\vec {B} = \frac {\vec {E}} {\vec {v}} = \frac {3.6e7} {7e8}} </math>

:Answer: <math> {\vec {B} = 0.051 T} </math>

The magnetic field is in the +z direction.

===Middling===

At a particular instant, a proton is moving with velocity <0,5e5,0> m/s and an electron is moving with velocity <-4.2e2,0,0> m/s. The electron is located 1.4e-3 m below the proton (in the -y direction). Determine the net force on the electron due to the proton.

:''Solution:''
:Step 1: <math> {\vec {F}_{net} = \vec {F}_{E} + \vec {F}_{B}} </math>
:Step 2: <math> {\vec {F}_{net} = \vec {F}_{E} + 0 = q\vec {E}} </math> (At the electron's location, <math> \vec B = 0 </math> because the velocity of the proton is parallel to <math> \hat{r} </math>)
:Step 3: <math> {\vec {E} = \frac {1} {4πεo} \frac {q} {r^2} \hat{r}} </math> (<math> \hat{r} = <0,-1,0> and r = 1.4e-3 m</math>)
:Step 4: <math> \vec E = <0,-7.35e-4,0> N/C </math>
:Step 5: <math> {\vec {F}_{net} = -e \vec E = <0,1.18e-22,0> N} </math>

===Difficult===
[[File:Middle_Example_A.jpg|thumb| '''Figure 5.''' Middle example diagram. ]]

A copper bar of length L and zero resistance slides at a constant velocity, v. There is a uniform magnetic field, B, directed into the page. A voltmeter is connected across a resistor, R, and reads ΔV. See '''Figure 5'''. Determine the direction of the magnetic force on the diagram and the current through the resistor. Your answer should be in terms of the given variables.

[[File:Middle_Example_B.jpg|thumb| '''Figure 6.''' Middle example solution. ]]

:''Solution:''
:Step 1: Direction of the Magnetic Force - Use right hand rule or <math> {\vec v \times\vec B} </math>. See '''Figure 6'''.
:Step 2: Current through the resistor -
::: <math> {ΔV}_{round trip} = 0 </math>
::: <math> {{ΔV}_{round trip} + motional emf - {ΔV}_{resistor} = 0} </math>
::: <math> motional emf = IR </math>
::: <math> I = \frac {motional emf} {R} </math>
::: <math> {|q\vec E| = |q\vec v\vec B|} </math>, <math> {|ΔV| = IR} </math>
::: <math> {qE = qvB} </math>
::: <math> E = vB </math>
::: <math> motional emf = |ΔV| = E*ΔL = vBL </math>
::: <math> {I = \frac {vBL} {R}} </math>

==Connectedness==

The Lorentz Force principle has been a component in many modern day inventions and critical building block for many physics principles. With known forces, we can predict the very important figure, the velocity and trajectory of a moving particle.

===Applications===

'''''Velocity Selector'''''
[[File:Velocityselector2.png|thumb| '''Figure 7.''' Illustration of a Velocity Selector ]]

The Velocity Selector is a device used to filter particles based on their velocity. A Velocity Selector uses controlled, perpendicular, electric and magnetic fields to filter certain charged particles (See '''Figure 7''' ). Particles with the correct speed will be unaffected while other particles will be deflected. This technique is used in technologies such as electron microscopes and spectrometers.

'''''Electric Motor'''''

An electric motor is a device that uses the Lorentz force to convert electric energy into mechanical energy. Using the [[magnetic torque]] principle, electric energy is created by using the magnetic field of a magnet. The [[torque]] laws are based off the principles of the net electric and magnetic forces.

===Other related topics===

Here are other principles that use the net force of magnetic and electric forces as a building block:

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

==History==

The relationship between electric and magnetic forces was first questioned in the mid-18th century when Johann Tobias Mayer (1760) and Henry Cavendish (1762) proposed that the force on magnetic poles and electrically charged objects followed the inverse-square law, which was proven to be true by Charles-Augustin in 1784.

Following Michael Faraday's proposal of the concept of electric and magnetic fields, James Clerk Maxwell was first to mathematically prove the concepts. In 1865, Maxwell's field equations consisted of some form of the Lorentz force equation, but at the time it was not clear how it related to forces on charged moving particles. J.J Thomson was the first to attempt a derivation of Maxwell's field equations, and he derived a basic form of the formula for the electromagnetic forces on a charged moving particle in relation to the properties of the particle and its external fields. In 1892, Hendrik Lorentz corrected mistakes of the old formula and derived the modern form of the equation, which contains the forces due to both electric and magnetic fields.

== See also ==

[[Hall Effect]]

[[Lorentz Force]]

[[Motional Emf]]

[[Electric Motors]]

[[Motional Emf using Faraday's Law]]

[[Inductance]]

===Further reading===

Books, Articles or other print media on this topic

[http://web.mit.edu/sahughes/www/8.022/lec10.pdf | MIT Physics notes on Lorentz force]

===External links===
Great youtube videos on Lorentz Force Law:
[https://www.youtube.com/watch?v=gINzRCOOs-8 |Lorentz Force Law Video 1]
[https://www.youtube.com/watch?v=PK6sEF9SNjw | Lorentz Force Law Video 2]

==References==

Boundless. “Electric vs. Magnetic Forces.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 05 Dec. 2015 from https://www.boundless.com/physics/textbooks/boundless-physics-textbook/magnetism-21/motion-of-a-charged-particle-in-a-magnetic-field-158/electric-vs-magnetic-forces-554-11176/

Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Vol. 2. Hoboken, NJ: Wiley, 2015. 812-814. Print.

All images found on google image search:
https://en.wikipedia.org/wiki/Magnetic_field
https://en.wikipedia.org/wiki/Wien_filter
http://aplusphysics.com/wordpress/regents/em/electric-field/

[[Fields]]

Electric Force

2018-03-20T21:20:22Z

Lpimentel3: /* A Computational Model */

'''CLAIMED BY Luis Pimentel Spring 2018'''

==The Main Idea==

There are two kinds of electric charge: positive and negative. Particles with like charges will repel each other, and particles with unlike charges attract each other. Calling a particle a "point particle" signifies an object with a radius so small in comparison to the distance between it and all other objects of interest. The object is then treated as if all its charge and mass were concentrated at a single point. Many times protons and electrons can be considered point particles due to its size.

The [[Electric Field]] created by a charge contains a force that is exerted on other charged particles or objects called an electric force. The strength of this electrical interaction is a vector quantity that has magnitude and direction. If the electric field at a particular location is known, then this field can be used to calculate the electric force of the particle being acted upon. The electric force is directly proportional to the amount of charge within each particle being acted upon by the other's electric field. Moreover, the magnitude of the force is inversely proportional to the square distance between the two interacting particles. It is important to remember that a particle cannot have an electric force on itself; there must be at least two interacting, charged components.

===A Mathematical Model===
The Coulomb Force Law
The formula for the magnitude of the electric force between two point charges is:

<math>F=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, where '''<math>{q}_{1}</math>''' and '''<math>{q}_{2}</math>''' are the magnitudes of electric charge of point 1 and point 2, and '''<math>r</math>''' is the distance between the two point charges. The units for electric force are in Newtons. The expression <math>\frac{1}{4 \pi \epsilon_0 }</math> is known as the electric constant and carries the value 9e9. Epsilon 0 defines electric permittivity of space.

Interestingly enough, one can see a relationship between this formula and the formula for gravitational force (<math>F={G} \frac{|{m}_{1}{m}_{2}|}{r^2} </math>). From this relationship, one can conclude that the interactions of two objects as a result of their charges or masses follow similar fundamental laws of physics.

'''Derivations of Electric Force'''

The electric force on a particle can also be written as:

<math>\vec F=q\vec E </math>, where '''<math>q</math>''' is the charge of the particle and '''<math>\vec E </math>''' is the external electric field.

This formula can be derived from <math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, the electric force between two point charges. The magnitude of the electric field created by a point charge is <math>|\vec E|=\frac{1}{4 \pi \epsilon_0 } \frac{|q|}{r^2} </math>, where '''<math>q</math>''' is the magnitude of the charge of the particle and '''<math>r</math>''' is the distance between the observation location and the point charge. Therefore, the magnitude of electric force between point charge 1 and point charge 2 can be written as:

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=|{q}_{2}|\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}|}{r^2}=|{q}_{2}||\vec{E}_{1}| </math>

The units of charge are in Coulombs and the units for electric field are in Newton/Coulombs, so this derivation is correct in its dimensions since multiplying the two units gives just Newtons. The Newton is the unit for electric force.

===A Computational Model===
'''Direction of the Electric Force'''

The electric force is along a straight line between the two point charges in the observed system. If the point charges have the same sign (i.e. both are either positively or negatively charged), then the charges repel each other. If the signs of the point charges are different (i.e. one is positively charged and one is negatively charged), then the point charges are attracted to each other. The electric force vector acts either in the same or opposite direction of the electric field acting on a particle, depending on the charge of that particle. Remember that negative charges attract, so the electric field of a negative charge will act from the observation location towards the charged negatively charged particle. If the observation particle is positively charged, then the electric force will act in the same direction as the electric field. If the observation particle is negatively charged, then the force will act in the opposite direction as the electric field. The same concepts apply to positive electric fields, which point away from the source location at the observation location.

[[File:ParticleAttration.jpg]]

Electric force follows Newton's third law of equal and opposite forces, meaning that the electric force experienced by one of the two interacting objects will be equal and opposite to the electric force of the first object.

----

A computational representation of the Electric force can be created using VPython.
The code below shows how we can find the net force, momentum and final position between two charged particles in VPython. This example uses two positively charged protons.

<code>
''''' #CONSTANTS '''''
E= 9e9 ''''' # Electric Force constant '''''
q1 = +1.6e-19 ''''' # Charge of proton 1'''''
q2 = +1.6e-19 ''''' # Charge of proton 2 '''''

while t < 200 :
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r = proton.pos - proton2.pos
rmag = mag(r)
rhat = r/rmag
Fnet = E*((q1*q2)/(mag2(r)))*rhat
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r2 = proton2.pos - proton.pos
rmag2 = mag(r2)
rhat2 = r2/rmag2
Fnet2 = E*((q1*q2)/(mag2(r2)))*rhat2
''''' # Update positions of BOTH protons '''''
pproton = pproton + Fnet*deltat
vproton = pproton/mproton
proton.pos = proton.pos + vproton*deltat
pproton2 = pproton2 + Fnet2*deltat
vproton2 = pproton2/mproton
proton2.pos = proton2.pos + vproton2*deltat
</code>

Notice that this iterative process is very similar to that working with [http://www.physicsbook.gatech.edu/Gravitational_Force Gravitational Forces].

Notice, however, that the direction of motion of each particle is not determined by relative position, but by the charges of the particles. The product of these charges ultimately determine whether the two particles will attract each other, or repel. This is demonstrated in the following simulations:

The trinket model linked demonstrates the Electric force interaction of a proton and an electron. [https://trinket.io/glowscript/c1cdb42527 Proton-Electron Electric Force Interaction]

The trinket model linked demonstrates the Electric force interaction between two protons. [https://trinket.io/glowscript/d5f7516fc2 Proton-Proton Electric Force Interaction]

==Examples==

===Simple===
'''Problem: '''Find the electric force of a -3 C particle in a region with an electric field of <math><7, 5, 0></math>N/C.

'''Step 1: '''Substitute values into the correct formula.

<math>\vec F=q\vec E </math>

<math>\vec F=(-3 C)<7, 5, 0></math>N/C

<math>\vec F=<-21, -15, 0></math>N

The electric force vector for this particle is <math><-21, -15, 0></math>N.

===Midding===
'''Problem: '''Find the magnitude of electric force on two charged particles located at <math> <0, 0, 0></math>m and <math> <0, 10, 0></math>m. The first particle has a charge of +5 nC and the second particle has a charge of -10 nC. Is the force attractive or repulsive?

'''Step 1: '''Find the distance between the two point charges.

<math>d=\sqrt{(0 m-0 m)^2+(0 m-10 m)^2+(0 m-0 m)^2}=\sqrt{100 m}=10 </math>m.

The distance between the two points is 10 m.

'''Step 2: '''Substitute values into the correct formula.

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=\frac{1}{4 \pi \epsilon_0 } \frac{|(5 nC)(-10 nC)|}{(10m)^2} </math>

<math>|\vec F|=4.5e-9 </math> N

The magnitude of electric force is <math>|\vec F|=4.5e-9 </math> N.

'''Step 3: '''Determine if force is attractive or repulsive.

Since the first particle is positively charged and the second is negatively charged, the force is attractive. The particles are attracted to each other.

===Difficult===
'''Problem:'''

[[File:SpringTest1Prob1.png]]

Using the graphic above, find a) the net force acting on particle -q' and b) the direction of the net force on this charge.

'''Step 1: '''Calculate the net force.

[[File:Force1.png]]
[[File:Force2.png]]

'''Step 2: '''Evaluate the direction of the force.

[[File:ForceDir.png]]

Problems involving electric force exclusively will not be more complicated than the above. However, the the electric force can be used in calculation of a net force acting on a particle in combination with non-Coulomb electric force and magnetic force.

==Connectedness==

Electric force is ubiquitous in everyday life, although it is not always evident. One common example of electric force is the attraction of clothes to one another after being washed. The charges caused by the machine-drying process create opposite, attractive charges on different pieces of clothing which cause them to stick together. Another example can be seen in refrigerator magnets due to the electromagnetic forces that enable magnetism. Thinking more complexly, the electric force is also prevalent in almost all forms of modern technology involving electricity. One particular example is the process of charging a smartphone: the electric force allows a current to be generated which transfers charge from outlets to the internal battery of these devices. A final, slightly more complicated example of the electric force is seen in the production of abrasive paper, whereby positively charged-smoothing particles are attracted to a negatively charged, smooth surface to create papers like sandpaper.

==History==

French physicist Charles-Augustin de Coulomb discovered in 1785 that the magnitude of electric force between two charged particles is directly proportional to the product of the absolute value of the two charges and inversely proportional to the distance squared between the two particles. He experimented with a torsion balance which consisted of an insulated bar suspended in the air by a silk thread. Coulomb attached a metal ball with a known charge to one end of the insulated bar. He then brought another ball with the same charge near the first ball. This distance between the two balls was recorded. The balls repelled each other, causing the silk thread to twist. The angle of the twist was measured and by knowing how much force was required for the thread to twist through the recorded angle, Coulomb was able to calculate the force between the two balls and derive the formula for electric force.

This [https://www.youtube.com/watch?v=FYSTGX-F1GM| video] explains Coulomb's experiment and the corresponding derivation of his law.

== See Also ==

Electric Field: [http://www.physicsbook.gatech.edu/Electric_Field]

Net Force: [http://www.physicsbook.gatech.edu/Lorentz_Force]

==References==

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

https://en.wikipedia.org/wiki/Coulomb's_law

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

http://www.jfinternational.com/ph/coulomb-law.html

[[Category:Forces]]

Electric Force

2018-03-20T04:27:51Z

Lpimentel3: /* A Computational Model */

'''CLAIMED BY Luis Pimentel Spring 2018'''

==The Main Idea==

There are two kinds of electric charge: positive and negative. Particles with like charges will repel each other, and particles with unlike charges attract each other. Calling a particle a "point particle" signifies an object with a radius so small in comparison to the distance between it and all other objects of interest. The object is then treated as if all its charge and mass were concentrated at a single point. Many times protons and electrons can be considered point particles due to its size.

The [[Electric Field]] created by a charge contains a force that is exerted on other charged particles or objects called an electric force. The strength of this electrical interaction is a vector quantity that has magnitude and direction. If the electric field at a particular location is known, then this field can be used to calculate the electric force of the particle being acted upon. The electric force is directly proportional to the amount of charge within each particle being acted upon by the other's electric field. Moreover, the magnitude of the force is inversely proportional to the square distance between the two interacting particles. It is important to remember that a particle cannot have an electric force on itself; there must be at least two interacting, charged components.

===A Mathematical Model===
The Coulomb Force Law
The formula for the magnitude of the electric force between two point charges is:

<math>F=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, where '''<math>{q}_{1}</math>''' and '''<math>{q}_{2}</math>''' are the magnitudes of electric charge of point 1 and point 2, and '''<math>r</math>''' is the distance between the two point charges. The units for electric force are in Newtons. The expression <math>\frac{1}{4 \pi \epsilon_0 }</math> is known as the electric constant and carries the value 9e9. Epsilon 0 defines electric permittivity of space.

Interestingly enough, one can see a relationship between this formula and the formula for gravitational force (<math>F={G} \frac{|{m}_{1}{m}_{2}|}{r^2} </math>). From this relationship, one can conclude that the interactions of two objects as a result of their charges or masses follow similar fundamental laws of physics.

'''Derivations of Electric Force'''

The electric force on a particle can also be written as:

<math>\vec F=q\vec E </math>, where '''<math>q</math>''' is the charge of the particle and '''<math>\vec E </math>''' is the external electric field.

This formula can be derived from <math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, the electric force between two point charges. The magnitude of the electric field created by a point charge is <math>|\vec E|=\frac{1}{4 \pi \epsilon_0 } \frac{|q|}{r^2} </math>, where '''<math>q</math>''' is the magnitude of the charge of the particle and '''<math>r</math>''' is the distance between the observation location and the point charge. Therefore, the magnitude of electric force between point charge 1 and point charge 2 can be written as:

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=|{q}_{2}|\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}|}{r^2}=|{q}_{2}||\vec{E}_{1}| </math>

The units of charge are in Coulombs and the units for electric field are in Newton/Coulombs, so this derivation is correct in its dimensions since multiplying the two units gives just Newtons. The Newton is the unit for electric force.

===A Computational Model===
'''Direction of the Electric Force'''

The electric force is along a straight line between the two point charges in the observed system. If the point charges have the same sign (i.e. both are either positively or negatively charged), then the charges repel each other. If the signs of the point charges are different (i.e. one is positively charged and one is negatively charged), then the point charges are attracted to each other. The electric force vector acts either in the same or opposite direction of the electric field acting on a particle, depending on the charge of that particle. Remember that negative charges attract, so the electric field of a negative charge will act from the observation location towards the charged negatively charged particle. If the observation particle is positively charged, then the electric force will act in the same direction as the electric field. If the observation particle is negatively charged, then the force will act in the opposite direction as the electric field. The same concepts apply to positive electric fields, which point away from the source location at the observation location.

[[File:ParticleAttration.jpg]]

Electric force follows Newton's third law of equal and opposite forces, meaning that the electric force experienced by one of the two interacting objects will be equal and opposite to the electric force of the first object.

----

A computational representation of the Electric force can be created using VPython.
The code below shows how we can find the net force, momentum and final position between two charged particles in VPython. This example uses two positively charged protons.

<code>
''''' #CONSTANTS '''''
E= 9e9 ''''' # Electric Force constant '''''
q1 = +1.6e-19 ''''' # Charge of proton 1'''''
q2 = +1.6e-19 ''''' # Charge of proton 2 '''''

while t < 200 :
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r = proton.pos - proton2.pos
rmag = mag(r)
rhat = r/rmag
Fnet = E*((q1*q2)/(mag2(r)))*rhat
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r2 = proton2.pos - proton.pos
rmag2 = mag(r2)
rhat2 = r2/rmag2
Fnet2 = E*((q1*q2)/(mag2(r2)))*rhat2
''''' # Update positions of BOTH protons '''''
pproton = pproton + Fnet*deltat
vproton = pproton/mproton
proton.pos = proton.pos + vproton*deltat
pproton2 = pproton2 + Fnet2*deltat
vproton2 = pproton2/mproton
proton2.pos = proton2.pos + vproton2*deltat
</code>

Notice that this iterative process is very similar to that working with [http://www.physicsbook.gatech.edu/Gravitational_Force Gravitational Forces].

Notice, however, that the motion of each particle is not determined by relative position, but by the charges of the particles. The product of these charges ultimately determine whether the two particles will attract each other, or repel. This is demonstrated in the following simulations:

The trinket model linked demonstrates the Electric force interaction of a proton and an electron. [https://trinket.io/glowscript/c1cdb42527 Proton-Electron Electric Force Interaction]

The trinket model linked demonstrates the Electric force interaction between two protons. [https://trinket.io/glowscript/d5f7516fc2 Proton-Proton Electric Force Interaction]

==Examples==

===Simple===
'''Problem: '''Find the electric force of a -3 C particle in a region with an electric field of <math><7, 5, 0></math>N/C.

'''Step 1: '''Substitute values into the correct formula.

<math>\vec F=q\vec E </math>

<math>\vec F=(-3 C)<7, 5, 0></math>N/C

<math>\vec F=<-21, -15, 0></math>N

The electric force vector for this particle is <math><-21, -15, 0></math>N.

===Midding===
'''Problem: '''Find the magnitude of electric force on two charged particles located at <math> <0, 0, 0></math>m and <math> <0, 10, 0></math>m. The first particle has a charge of +5 nC and the second particle has a charge of -10 nC. Is the force attractive or repulsive?

'''Step 1: '''Find the distance between the two point charges.

<math>d=\sqrt{(0 m-0 m)^2+(0 m-10 m)^2+(0 m-0 m)^2}=\sqrt{100 m}=10 </math>m.

The distance between the two points is 10 m.

'''Step 2: '''Substitute values into the correct formula.

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=\frac{1}{4 \pi \epsilon_0 } \frac{|(5 nC)(-10 nC)|}{(10m)^2} </math>

<math>|\vec F|=4.5e-9 </math> N

The magnitude of electric force is <math>|\vec F|=4.5e-9 </math> N.

'''Step 3: '''Determine if force is attractive or repulsive.

Since the first particle is positively charged and the second is negatively charged, the force is attractive. The particles are attracted to each other.

===Difficult===
'''Problem:'''

[[File:SpringTest1Prob1.png]]

Using the graphic above, find a) the net force acting on particle -q' and b) the direction of the net force on this charge.

'''Step 1: '''Calculate the net force.

[[File:Force1.png]]
[[File:Force2.png]]

'''Step 2: '''Evaluate the direction of the force.

[[File:ForceDir.png]]

Problems involving electric force exclusively will not be more complicated than the above. However, the the electric force can be used in calculation of a net force acting on a particle in combination with non-Coulomb electric force and magnetic force.

==Connectedness==

Electric force is ubiquitous in everyday life, although it is not always evident. One common example of electric force is the attraction of clothes to one another after being washed. The charges caused by the machine-drying process create opposite, attractive charges on different pieces of clothing which cause them to stick together. Another example can be seen in refrigerator magnets due to the electromagnetic forces that enable magnetism. Thinking more complexly, the electric force is also prevalent in almost all forms of modern technology involving electricity. One particular example is the process of charging a smartphone: the electric force allows a current to be generated which transfers charge from outlets to the internal battery of these devices. A final, slightly more complicated example of the electric force is seen in the production of abrasive paper, whereby positively charged-smoothing particles are attracted to a negatively charged, smooth surface to create papers like sandpaper.

==History==

French physicist Charles-Augustin de Coulomb discovered in 1785 that the magnitude of electric force between two charged particles is directly proportional to the product of the absolute value of the two charges and inversely proportional to the distance squared between the two particles. He experimented with a torsion balance which consisted of an insulated bar suspended in the air by a silk thread. Coulomb attached a metal ball with a known charge to one end of the insulated bar. He then brought another ball with the same charge near the first ball. This distance between the two balls was recorded. The balls repelled each other, causing the silk thread to twist. The angle of the twist was measured and by knowing how much force was required for the thread to twist through the recorded angle, Coulomb was able to calculate the force between the two balls and derive the formula for electric force.

This [https://www.youtube.com/watch?v=FYSTGX-F1GM| video] explains Coulomb's experiment and the corresponding derivation of his law.

== See Also ==

Electric Field: [http://www.physicsbook.gatech.edu/Electric_Field]

Net Force: [http://www.physicsbook.gatech.edu/Lorentz_Force]

==References==

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

https://en.wikipedia.org/wiki/Coulomb's_law

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

http://www.jfinternational.com/ph/coulomb-law.html

[[Category:Forces]]

Electric Force

2018-03-20T04:25:26Z

Lpimentel3: /* A Computational Model */

'''CLAIMED BY Luis Pimentel Spring 2018'''

==The Main Idea==

There are two kinds of electric charge: positive and negative. Particles with like charges will repel each other, and particles with unlike charges attract each other. Calling a particle a "point particle" signifies an object with a radius so small in comparison to the distance between it and all other objects of interest. The object is then treated as if all its charge and mass were concentrated at a single point. Many times protons and electrons can be considered point particles due to its size.

The [[Electric Field]] created by a charge contains a force that is exerted on other charged particles or objects called an electric force. The strength of this electrical interaction is a vector quantity that has magnitude and direction. If the electric field at a particular location is known, then this field can be used to calculate the electric force of the particle being acted upon. The electric force is directly proportional to the amount of charge within each particle being acted upon by the other's electric field. Moreover, the magnitude of the force is inversely proportional to the square distance between the two interacting particles. It is important to remember that a particle cannot have an electric force on itself; there must be at least two interacting, charged components.

===A Mathematical Model===
The Coulomb Force Law
The formula for the magnitude of the electric force between two point charges is:

<math>F=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, where '''<math>{q}_{1}</math>''' and '''<math>{q}_{2}</math>''' are the magnitudes of electric charge of point 1 and point 2, and '''<math>r</math>''' is the distance between the two point charges. The units for electric force are in Newtons. The expression <math>\frac{1}{4 \pi \epsilon_0 }</math> is known as the electric constant and carries the value 9e9. Epsilon 0 defines electric permittivity of space.

Interestingly enough, one can see a relationship between this formula and the formula for gravitational force (<math>F={G} \frac{|{m}_{1}{m}_{2}|}{r^2} </math>). From this relationship, one can conclude that the interactions of two objects as a result of their charges or masses follow similar fundamental laws of physics.

'''Derivations of Electric Force'''

The electric force on a particle can also be written as:

<math>\vec F=q\vec E </math>, where '''<math>q</math>''' is the charge of the particle and '''<math>\vec E </math>''' is the external electric field.

This formula can be derived from <math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, the electric force between two point charges. The magnitude of the electric field created by a point charge is <math>|\vec E|=\frac{1}{4 \pi \epsilon_0 } \frac{|q|}{r^2} </math>, where '''<math>q</math>''' is the magnitude of the charge of the particle and '''<math>r</math>''' is the distance between the observation location and the point charge. Therefore, the magnitude of electric force between point charge 1 and point charge 2 can be written as:

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=|{q}_{2}|\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}|}{r^2}=|{q}_{2}||\vec{E}_{1}| </math>

The units of charge are in Coulombs and the units for electric field are in Newton/Coulombs, so this derivation is correct in its dimensions since multiplying the two units gives just Newtons. The Newton is the unit for electric force.

===A Computational Model===
'''Direction of the Electric Force'''

The electric force is along a straight line between the two point charges in the observed system. If the point charges have the same sign (i.e. both are either positively or negatively charged), then the charges repel each other. If the signs of the point charges are different (i.e. one is positively charged and one is negatively charged), then the point charges are attracted to each other. The electric force vector acts either in the same or opposite direction of the electric field acting on a particle, depending on the charge of that particle. Remember that negative charges attract, so the electric field of a negative charge will act from the observation location towards the charged negatively charged particle. If the observation particle is positively charged, then the electric force will act in the same direction as the electric field. If the observation particle is negatively charged, then the force will act in the opposite direction as the electric field. The same concepts apply to positive electric fields, which point away from the source location at the observation location.

[[File:ParticleAttration.jpg]]

Electric force follows Newton's third law of equal and opposite forces, meaning that the electric force experienced by one of the two interacting objects will be equal and opposite to the electric force of the first object.

----

A computational representation of the Electric force can be created using VPython.
The code below shows how we can find the net force, momentum and final position between two charged particles in VPython. This example uses two positively charged protons.

<code>
''''' #CONSTANTS '''''
E= 9e9 ''''' # Electric Force constant '''''
q1 = +1.6e-19 ''''' # Charge of proton 1'''''
q2 = +1.6e-19 ''''' # Charge of proton 2 '''''

while t < 200 :
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r = proton.pos - proton2.pos
rmag = mag(r)
rhat = r/rmag
Fnet = E*((q1*q2)/(mag2(r)))*rhat
''''' # Calculate electric force acting on proton 1 by proton 2 '''''
r2 = proton2.pos - proton.pos
rmag2 = mag(r2)
rhat2 = r2/rmag2
Fnet2 = E*((q1*q2)/(mag2(r2)))*rhat2
''''' # Update positions of BOTH protons '''''
pproton = pproton + Fnet*deltat
vproton = pproton/mproton
proton.pos = proton.pos + vproton*deltat
pproton2 = pproton2 + Fnet2*deltat
vproton2 = pproton2/mproton
proton2.pos = proton2.pos + vproton2*deltat

Notice that this iterative process is very similar to that working with [http://www.physicsbook.gatech.edu/Gravitational_Force Gravitational Forces].

Notice, however, that the motion of each particle is not determined by relative position, but by the charges of the particles. The product of these charges ultimately determine whether the two particles will attract each other, or repel. This is demonstrated in the following simulations:

The trinket model linked demonstrates the Electric force interaction of a proton and an electron. [https://trinket.io/glowscript/c1cdb42527 Proton-Electron Electric Force Interaction]

The trinket model linked demonstrates the Electric force interaction between two protons. [https://trinket.io/glowscript/d5f7516fc2 Proton-Proton Electric Force Interaction]

==Examples==

===Simple===
'''Problem: '''Find the electric force of a -3 C particle in a region with an electric field of <math><7, 5, 0></math>N/C.

'''Step 1: '''Substitute values into the correct formula.

<math>\vec F=q\vec E </math>

<math>\vec F=(-3 C)<7, 5, 0></math>N/C

<math>\vec F=<-21, -15, 0></math>N

The electric force vector for this particle is <math><-21, -15, 0></math>N.

===Midding===
'''Problem: '''Find the magnitude of electric force on two charged particles located at <math> <0, 0, 0></math>m and <math> <0, 10, 0></math>m. The first particle has a charge of +5 nC and the second particle has a charge of -10 nC. Is the force attractive or repulsive?

'''Step 1: '''Find the distance between the two point charges.

<math>d=\sqrt{(0 m-0 m)^2+(0 m-10 m)^2+(0 m-0 m)^2}=\sqrt{100 m}=10 </math>m.

The distance between the two points is 10 m.

'''Step 2: '''Substitute values into the correct formula.

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=\frac{1}{4 \pi \epsilon_0 } \frac{|(5 nC)(-10 nC)|}{(10m)^2} </math>

<math>|\vec F|=4.5e-9 </math> N

The magnitude of electric force is <math>|\vec F|=4.5e-9 </math> N.

'''Step 3: '''Determine if force is attractive or repulsive.

Since the first particle is positively charged and the second is negatively charged, the force is attractive. The particles are attracted to each other.

===Difficult===
'''Problem:'''

[[File:SpringTest1Prob1.png]]

Using the graphic above, find a) the net force acting on particle -q' and b) the direction of the net force on this charge.

'''Step 1: '''Calculate the net force.

[[File:Force1.png]]
[[File:Force2.png]]

'''Step 2: '''Evaluate the direction of the force.

[[File:ForceDir.png]]

Problems involving electric force exclusively will not be more complicated than the above. However, the the electric force can be used in calculation of a net force acting on a particle in combination with non-Coulomb electric force and magnetic force.

==Connectedness==

Electric force is ubiquitous in everyday life, although it is not always evident. One common example of electric force is the attraction of clothes to one another after being washed. The charges caused by the machine-drying process create opposite, attractive charges on different pieces of clothing which cause them to stick together. Another example can be seen in refrigerator magnets due to the electromagnetic forces that enable magnetism. Thinking more complexly, the electric force is also prevalent in almost all forms of modern technology involving electricity. One particular example is the process of charging a smartphone: the electric force allows a current to be generated which transfers charge from outlets to the internal battery of these devices. A final, slightly more complicated example of the electric force is seen in the production of abrasive paper, whereby positively charged-smoothing particles are attracted to a negatively charged, smooth surface to create papers like sandpaper.

==History==

French physicist Charles-Augustin de Coulomb discovered in 1785 that the magnitude of electric force between two charged particles is directly proportional to the product of the absolute value of the two charges and inversely proportional to the distance squared between the two particles. He experimented with a torsion balance which consisted of an insulated bar suspended in the air by a silk thread. Coulomb attached a metal ball with a known charge to one end of the insulated bar. He then brought another ball with the same charge near the first ball. This distance between the two balls was recorded. The balls repelled each other, causing the silk thread to twist. The angle of the twist was measured and by knowing how much force was required for the thread to twist through the recorded angle, Coulomb was able to calculate the force between the two balls and derive the formula for electric force.

This [https://www.youtube.com/watch?v=FYSTGX-F1GM| video] explains Coulomb's experiment and the corresponding derivation of his law.

== See Also ==

Electric Field: [http://www.physicsbook.gatech.edu/Electric_Field]

Net Force: [http://www.physicsbook.gatech.edu/Lorentz_Force]

==References==

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

https://en.wikipedia.org/wiki/Coulomb's_law

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

http://www.jfinternational.com/ph/coulomb-law.html

[[Category:Forces]]

Electric Force

2018-03-20T04:10:50Z

Lpimentel3:

'''CLAIMED BY Luis Pimentel Spring 2018'''

==The Main Idea==

There are two kinds of electric charge: positive and negative. Particles with like charges will repel each other, and particles with unlike charges attract each other. Calling a particle a "point particle" signifies an object with a radius so small in comparison to the distance between it and all other objects of interest. The object is then treated as if all its charge and mass were concentrated at a single point. Many times protons and electrons can be considered point particles due to its size.

The [[Electric Field]] created by a charge contains a force that is exerted on other charged particles or objects called an electric force. The strength of this electrical interaction is a vector quantity that has magnitude and direction. If the electric field at a particular location is known, then this field can be used to calculate the electric force of the particle being acted upon. The electric force is directly proportional to the amount of charge within each particle being acted upon by the other's electric field. Moreover, the magnitude of the force is inversely proportional to the square distance between the two interacting particles. It is important to remember that a particle cannot have an electric force on itself; there must be at least two interacting, charged components.

===A Mathematical Model===
The Coulomb Force Law
The formula for the magnitude of the electric force between two point charges is:

<math>F=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, where '''<math>{q}_{1}</math>''' and '''<math>{q}_{2}</math>''' are the magnitudes of electric charge of point 1 and point 2, and '''<math>r</math>''' is the distance between the two point charges. The units for electric force are in Newtons. The expression <math>\frac{1}{4 \pi \epsilon_0 }</math> is known as the electric constant and carries the value 9e9. Epsilon 0 defines electric permittivity of space.

Interestingly enough, one can see a relationship between this formula and the formula for gravitational force (<math>F={G} \frac{|{m}_{1}{m}_{2}|}{r^2} </math>). From this relationship, one can conclude that the interactions of two objects as a result of their charges or masses follow similar fundamental laws of physics.

'''Derivations of Electric Force'''

The electric force on a particle can also be written as:

<math>\vec F=q\vec E </math>, where '''<math>q</math>''' is the charge of the particle and '''<math>\vec E </math>''' is the external electric field.

This formula can be derived from <math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2} </math>, the electric force between two point charges. The magnitude of the electric field created by a point charge is <math>|\vec E|=\frac{1}{4 \pi \epsilon_0 } \frac{|q|}{r^2} </math>, where '''<math>q</math>''' is the magnitude of the charge of the particle and '''<math>r</math>''' is the distance between the observation location and the point charge. Therefore, the magnitude of electric force between point charge 1 and point charge 2 can be written as:

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=|{q}_{2}|\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}|}{r^2}=|{q}_{2}||\vec{E}_{1}| </math>

The units of charge are in Coulombs and the units for electric field are in Newton/Coulombs, so this derivation is correct in its dimensions since multiplying the two units gives just Newtons. The Newton is the unit for electric force.

===A Computational Model===
'''Direction of the Electric Force'''

The electric force is along a straight line between the two point charges in the observed system. If the point charges have the same sign (i.e. both are either positively or negatively charged), then the charges repel each other. If the signs of the point charges are different (i.e. one is positively charged and one is negatively charged), then the point charges are attracted to each other. The electric force vector acts either in the same or opposite direction of the electric field acting on a particle, depending on the charge of that particle. Remember that negative charges attract, so the electric field of a negative charge will act from the observation location towards the charged negatively charged particle. If the observation particle is positively charged, then the electric force will act in the same direction as the electric field. If the observation particle is negatively charged, then the force will act in the opposite direction as the electric field. The same concepts apply to positive electric fields, which point away from the source location at the observation location.

[[File:ParticleAttration.jpg]]

Electric force follows Newton's third law of equal and opposite forces, meaning that the electric force experienced by one of the two interacting objects will be equal and opposite to the electric force of the first object.
==Examples==

===Simple===
'''Problem: '''Find the electric force of a -3 C particle in a region with an electric field of <math><7, 5, 0></math>N/C.

'''Step 1: '''Substitute values into the correct formula.

<math>\vec F=q\vec E </math>

<math>\vec F=(-3 C)<7, 5, 0></math>N/C

<math>\vec F=<-21, -15, 0></math>N

The electric force vector for this particle is <math><-21, -15, 0></math>N.

===Midding===
'''Problem: '''Find the magnitude of electric force on two charged particles located at <math> <0, 0, 0></math>m and <math> <0, 10, 0></math>m. The first particle has a charge of +5 nC and the second particle has a charge of -10 nC. Is the force attractive or repulsive?

'''Step 1: '''Find the distance between the two point charges.

<math>d=\sqrt{(0 m-0 m)^2+(0 m-10 m)^2+(0 m-0 m)^2}=\sqrt{100 m}=10 </math>m.

The distance between the two points is 10 m.

'''Step 2: '''Substitute values into the correct formula.

<math>|\vec F|=\frac{1}{4 \pi \epsilon_0 } \frac{|{q}_{1}{q}_{2}|}{r^2}=\frac{1}{4 \pi \epsilon_0 } \frac{|(5 nC)(-10 nC)|}{(10m)^2} </math>

<math>|\vec F|=4.5e-9 </math> N

The magnitude of electric force is <math>|\vec F|=4.5e-9 </math> N.

'''Step 3: '''Determine if force is attractive or repulsive.

Since the first particle is positively charged and the second is negatively charged, the force is attractive. The particles are attracted to each other.

===Difficult===
'''Problem:'''

[[File:SpringTest1Prob1.png]]

Using the graphic above, find a) the net force acting on particle -q' and b) the direction of the net force on this charge.

'''Step 1: '''Calculate the net force.

[[File:Force1.png]]
[[File:Force2.png]]

'''Step 2: '''Evaluate the direction of the force.

[[File:ForceDir.png]]

Problems involving electric force exclusively will not be more complicated than the above. However, the the electric force can be used in calculation of a net force acting on a particle in combination with non-Coulomb electric force and magnetic force.

==Connectedness==

Electric force is ubiquitous in everyday life, although it is not always evident. One common example of electric force is the attraction of clothes to one another after being washed. The charges caused by the machine-drying process create opposite, attractive charges on different pieces of clothing which cause them to stick together. Another example can be seen in refrigerator magnets due to the electromagnetic forces that enable magnetism. Thinking more complexly, the electric force is also prevalent in almost all forms of modern technology involving electricity. One particular example is the process of charging a smartphone: the electric force allows a current to be generated which transfers charge from outlets to the internal battery of these devices. A final, slightly more complicated example of the electric force is seen in the production of abrasive paper, whereby positively charged-smoothing particles are attracted to a negatively charged, smooth surface to create papers like sandpaper.

==History==

French physicist Charles-Augustin de Coulomb discovered in 1785 that the magnitude of electric force between two charged particles is directly proportional to the product of the absolute value of the two charges and inversely proportional to the distance squared between the two particles. He experimented with a torsion balance which consisted of an insulated bar suspended in the air by a silk thread. Coulomb attached a metal ball with a known charge to one end of the insulated bar. He then brought another ball with the same charge near the first ball. This distance between the two balls was recorded. The balls repelled each other, causing the silk thread to twist. The angle of the twist was measured and by knowing how much force was required for the thread to twist through the recorded angle, Coulomb was able to calculate the force between the two balls and derive the formula for electric force.

This [https://www.youtube.com/watch?v=FYSTGX-F1GM| video] explains Coulomb's experiment and the corresponding derivation of his law.

== See Also ==

Electric Field: [http://www.physicsbook.gatech.edu/Electric_Field]

Net Force: [http://www.physicsbook.gatech.edu/Lorentz_Force]

==References==

Matter & Interactions, Vol. II: Electric and Magnetic Interactions, 4th Edition

https://en.wikipedia.org/wiki/Coulomb's_law

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

http://www.jfinternational.com/ph/coulomb-law.html

[[Category:Forces]]