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1.5 Other Charge Distributions - Fields & Potentials

4 min readdecember 29, 2022

Peter Apps

Peter Apps

Peter Apps

Peter Apps

This topic is a catch-all for more advanced applications of , , and calculating . All of the concepts are covered in the previous sections, so if you need to, re-read those first.

Extended Charge Distributions

Up until now, we've dealt with charged objects as points, but sometimes we can't approximate the charge to be in a single location. These extended are cases where the charge is in a ring, or line, or sheet.

The proccess for tackling all of these are very similar. We're going to chunk the total charge Q into into pieces, dq, which each contributes a partial field, dE. The entire equation is dE = k dq/r^2 * r, where r is the . By integrating that equation, we can sum up all of the dq which sums all of the dE to find the total field E.

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-v1S5CQhDau3k.png?alt=media&token=8a9246ee-4b4f-4d94-a583-331e3d8cee7f

Image from dev.physicslab.org

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-SMay258Zr2uS.PNG?alt=media&token=615e16ba-114e-4491-8380-e9a39a4d2da3

If we assume that x >>> a, then we get the same result as for a point charge we found in Section 1. 2! This is a more general case, but we can see that it simplifies to the basic case!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-AlNmUcLNZGYY.PNG?alt=media&token=441ed930-cba3-41c7-8526-78ad7a288e3b

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DZIdQkMSeBWk.png?alt=media&token=706d2595-1450-47ac-b256-186c02846169

Image created by author

Start by finding dq in terms of y using linear charge density ƛ.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.16-UFZF8nR4VHQA.png?alt=media&token=88f316df-71e8-4eec-8e82-8152c693529f

Then, plug into our integral just like in the previous example. In this case, we can find r by using the pythagoreon theorem between y and x.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.17-z8tEQjtcZMkl.png?alt=media&token=9179139e-ce75-43ac-8825-5346f13ec882

Gauss' Law for Various Shapes

You should be able to use to derive these if you need to. Check out hyperphysics for more info.

:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Beuigbr2OzdR.PNG?alt=media&token=7545bab0-b520-4248-9063-fc03ccb94a60

Imagine enclosing the in a cylinder, use linear charge density λ = Q/L to find the enclosed charge, and then use A=2πrL.

Point, Hoop, or Sphere (fully enclosed):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.19-Ko9vubD7YVa2.png?alt=media&token=4abb450d-01dd-4bcf-bf48-9a523a3ba3c8

Imagine wrapping the point or hoop in a sphere. Then use A = 4πr^2.

Sphere (not fully enclosed):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-804RIXsSlmd5.PNG?alt=media&token=449fd4eb-d21a-4cb1-bd27-e3c8f54c70f3

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-anVgVuuEidYQ.jpg_revision%3D1?alt=media&token=2be99a40-e51c-4ccf-bbea-443280030676

Image Courtesy of phys.libretexts.org (CC BY 4.0)

Insulating Sheet of Charge:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.20-EZQ9NwtDlnKS.png?alt=media&token=7f9fe5b9-6290-4390-be96-3dfda8e295eb

Enclose the sheet in a rectangle, being careful because there are fields on both side of the sheet. Use area charge density σ = Q/A --> q_enc = σA and E∲dA=2EA.

Potential Difference for a Variety of Shapes

This is more straightforward. For all the shapes, simply apply to find an expression for E, then plug that into

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-oN911m1hneBB.PNG?alt=media&token=9ab991ed-b2af-4906-b19d-4313c74d8e7a

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-HJ00jtXtQXEB.PNG?alt=media&token=7acbc9ae-2c58-4a68-a59f-b00be995601b

Example (moving from the sheet to a distance d away):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-hZxflwENo5Nr.PNG?alt=media&token=ee659f1f-750d-4c1d-b971-873cc9665e76

Practice Questions

1.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-i8OAyI6TDfgS.png?alt=media&token=74d7f969-8851-459a-9672-11ff53f1c8da

a)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-lakjQHPDKgTX.png?alt=media&token=edbae3cb-cadf-4f47-80c9-afb164fa4100

b)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-xL3MBxXBnFn6.png?alt=media&token=438fbb45-5c02-404b-bd49-7607e3e427e4

Image from collegeboard.org

Answer:

(a) The shells are conducting so the charge is only on their surfaces. Since we're in the region between r1 and r2, we fully enclose Q1. Using , A is the correct answer.

(b) Relative to infinity, the total at r2 is V = k(Q1 + Q2)/r2, when we're between the sphere, we no longer have a change due to V_Q2 because the is CONSTANT inside of a conductor (while the Electric Field is zero -- this is an important thing to remember for the AP exam!). However, V_Q1 still varies according to 1/r. This means that E will be the correct answer.

2.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jgorktM8p9fU.png?alt=media&token=91396016-cbee-47ff-9a08-914ddb0bdc3c

Image from collegeboard.org

Answer:

Choice B Is correct. Every point is equidistant from a given x value so V = kQ/r where r =√x^2 + a^2

3.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fquestion%203-qxhPHAVsLQLK.png?alt=media&token=253a75a7-d63c-4bb8-a31c-37a5556ca85d

Image from collegeboard.org

Answer:

D is correct. Outside the spheres E = 0 since the total charge enclosed is 0. Inside the negative shell, the looks like a positively charged sphere (constant when r < a, and proportional to 1/r when r > a).

Practice FRQ

(courtesy of the College Board website, AP Physics C: Electricity + Magnetism 2007 Free Response Section, Question #2)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.27-3lRLDDqcEi8L.png?alt=media&token=10ee2c35-404d-4a8b-8f32-c933bad7c6cc

In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a charge -Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants.

(a) Using Gauss’s law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions.

i. Within the solid sphere (r < a)

ii. Between the solid sphere and the spherical shell (a < r < 2a)

iii. Within the spherical shell (2a < r < 3a)

iv. Outside the spherical shell (r > 3a)

(b) What is the electric at the outer surface of the spherical shell (r = 3a)? Explain your reasoning.

(c) Derive an expression for the between points X and Y shown in the figure.

Scoring Guidelines

The scoring guidelines are located at this PDF from the College Board website for Question #2.

Key Terms to Review (13)

Area charge density (σ)

: Area charge density refers to the amount of electric charge per unit area. It is a measure of how much charge is distributed over a given surface.

Charge Distributions

: Charge distributions refer to the arrangement of electric charges in a given space. It describes how charges are distributed and spread out within an object or system.

Conducting Sheets

: Conducting sheets are materials that allow charges to move freely within them due to their high conductivity. They can redistribute charges on their surface and shield external electric fields.

Coulomb's constant

: Coulomb's constant, denoted as k, is a proportionality constant that relates the electrostatic force between two charged objects to their charges and the distance between them.

Electric Field is zero inside a conductor

: Inside a conductor, the electric field is always zero. This means that there is no net force acting on charges inside a conductor, and they are in electrostatic equilibrium.

Electric potential difference V_X - V_Y

: The electric potential difference between two points, V_X and V_Y, is the change in electric potential energy per unit charge as a charge moves from point X to point Y.

Electric Potentials

: Electric potentials refer to the amount of electrical potential energy per unit charge at any given point in an electric field. It represents how much work is required to move a positive test charge from infinity to that point without acceleration.

Gauss' Law

: Gauss' Law relates the electric flux through a closed surface to the total charge enclosed by that surface. It provides a convenient way to calculate electric fields for symmetric charge distributions.

Line of Charge

: A line of charge refers to a distribution of electric charge that is concentrated along an infinitely long, straight line.

Potential

: The potential refers to the electric potential energy per unit charge at a specific point in an electric field. It represents the work done to move a positive test charge from infinity to that point.

Potential Difference

: Potential difference, also known as voltage, is the difference in electric potential energy per unit charge between two points in an electric circuit. It represents how much work is done on each unit of charge when it moves from one point to another.

radius vector

: In physics, radius vector (r) refers to a vector that represents both magnitude and direction from an origin point to another point in space.

Ring of Charge

: A ring of charge refers to a circular loop or disc with uniformly distributed charge along its circumference.

1.5 Other Charge Distributions - Fields & Potentials

4 min readdecember 29, 2022

Peter Apps

Peter Apps

Peter Apps

Peter Apps

This topic is a catch-all for more advanced applications of , , and calculating . All of the concepts are covered in the previous sections, so if you need to, re-read those first.

Extended Charge Distributions

Up until now, we've dealt with charged objects as points, but sometimes we can't approximate the charge to be in a single location. These extended are cases where the charge is in a ring, or line, or sheet.

The proccess for tackling all of these are very similar. We're going to chunk the total charge Q into into pieces, dq, which each contributes a partial field, dE. The entire equation is dE = k dq/r^2 * r, where r is the . By integrating that equation, we can sum up all of the dq which sums all of the dE to find the total field E.

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-v1S5CQhDau3k.png?alt=media&token=8a9246ee-4b4f-4d94-a583-331e3d8cee7f

Image from dev.physicslab.org

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-SMay258Zr2uS.PNG?alt=media&token=615e16ba-114e-4491-8380-e9a39a4d2da3

If we assume that x >>> a, then we get the same result as for a point charge we found in Section 1. 2! This is a more general case, but we can see that it simplifies to the basic case!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-AlNmUcLNZGYY.PNG?alt=media&token=441ed930-cba3-41c7-8526-78ad7a288e3b

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DZIdQkMSeBWk.png?alt=media&token=706d2595-1450-47ac-b256-186c02846169

Image created by author

Start by finding dq in terms of y using linear charge density ƛ.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.16-UFZF8nR4VHQA.png?alt=media&token=88f316df-71e8-4eec-8e82-8152c693529f

Then, plug into our integral just like in the previous example. In this case, we can find r by using the pythagoreon theorem between y and x.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.17-z8tEQjtcZMkl.png?alt=media&token=9179139e-ce75-43ac-8825-5346f13ec882

Gauss' Law for Various Shapes

You should be able to use to derive these if you need to. Check out hyperphysics for more info.

:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Beuigbr2OzdR.PNG?alt=media&token=7545bab0-b520-4248-9063-fc03ccb94a60

Imagine enclosing the in a cylinder, use linear charge density λ = Q/L to find the enclosed charge, and then use A=2πrL.

Point, Hoop, or Sphere (fully enclosed):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.19-Ko9vubD7YVa2.png?alt=media&token=4abb450d-01dd-4bcf-bf48-9a523a3ba3c8

Imagine wrapping the point or hoop in a sphere. Then use A = 4πr^2.

Sphere (not fully enclosed):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-804RIXsSlmd5.PNG?alt=media&token=449fd4eb-d21a-4cb1-bd27-e3c8f54c70f3

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-anVgVuuEidYQ.jpg_revision%3D1?alt=media&token=2be99a40-e51c-4ccf-bbea-443280030676

Image Courtesy of phys.libretexts.org (CC BY 4.0)

Insulating Sheet of Charge:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.20-EZQ9NwtDlnKS.png?alt=media&token=7f9fe5b9-6290-4390-be96-3dfda8e295eb

Enclose the sheet in a rectangle, being careful because there are fields on both side of the sheet. Use area charge density σ = Q/A --> q_enc = σA and E∲dA=2EA.

Potential Difference for a Variety of Shapes

This is more straightforward. For all the shapes, simply apply to find an expression for E, then plug that into

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-oN911m1hneBB.PNG?alt=media&token=9ab991ed-b2af-4906-b19d-4313c74d8e7a

Example:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-HJ00jtXtQXEB.PNG?alt=media&token=7acbc9ae-2c58-4a68-a59f-b00be995601b

Example (moving from the sheet to a distance d away):

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-hZxflwENo5Nr.PNG?alt=media&token=ee659f1f-750d-4c1d-b971-873cc9665e76

Practice Questions

1.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-i8OAyI6TDfgS.png?alt=media&token=74d7f969-8851-459a-9672-11ff53f1c8da

a)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-lakjQHPDKgTX.png?alt=media&token=edbae3cb-cadf-4f47-80c9-afb164fa4100

b)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-xL3MBxXBnFn6.png?alt=media&token=438fbb45-5c02-404b-bd49-7607e3e427e4

Image from collegeboard.org

Answer:

(a) The shells are conducting so the charge is only on their surfaces. Since we're in the region between r1 and r2, we fully enclose Q1. Using , A is the correct answer.

(b) Relative to infinity, the total at r2 is V = k(Q1 + Q2)/r2, when we're between the sphere, we no longer have a change due to V_Q2 because the is CONSTANT inside of a conductor (while the Electric Field is zero -- this is an important thing to remember for the AP exam!). However, V_Q1 still varies according to 1/r. This means that E will be the correct answer.

2.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jgorktM8p9fU.png?alt=media&token=91396016-cbee-47ff-9a08-914ddb0bdc3c

Image from collegeboard.org

Answer:

Choice B Is correct. Every point is equidistant from a given x value so V = kQ/r where r =√x^2 + a^2

3.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fquestion%203-qxhPHAVsLQLK.png?alt=media&token=253a75a7-d63c-4bb8-a31c-37a5556ca85d

Image from collegeboard.org

Answer:

D is correct. Outside the spheres E = 0 since the total charge enclosed is 0. Inside the negative shell, the looks like a positively charged sphere (constant when r < a, and proportional to 1/r when r > a).

Practice FRQ

(courtesy of the College Board website, AP Physics C: Electricity + Magnetism 2007 Free Response Section, Question #2)

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%202022-12-29%20at%202.27-3lRLDDqcEi8L.png?alt=media&token=10ee2c35-404d-4a8b-8f32-c933bad7c6cc

In the figure above, a nonconducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a nonconducting spherical shell of inner radius 2a and outer radius 3a that has a charge -Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants.

(a) Using Gauss’s law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions.

i. Within the solid sphere (r < a)

ii. Between the solid sphere and the spherical shell (a < r < 2a)

iii. Within the spherical shell (2a < r < 3a)

iv. Outside the spherical shell (r > 3a)

(b) What is the electric at the outer surface of the spherical shell (r = 3a)? Explain your reasoning.

(c) Derive an expression for the between points X and Y shown in the figure.

Scoring Guidelines

The scoring guidelines are located at this PDF from the College Board website for Question #2.

Key Terms to Review (13)

Area charge density (σ)

: Area charge density refers to the amount of electric charge per unit area. It is a measure of how much charge is distributed over a given surface.

Charge Distributions

: Charge distributions refer to the arrangement of electric charges in a given space. It describes how charges are distributed and spread out within an object or system.

Conducting Sheets

: Conducting sheets are materials that allow charges to move freely within them due to their high conductivity. They can redistribute charges on their surface and shield external electric fields.

Coulomb's constant

: Coulomb's constant, denoted as k, is a proportionality constant that relates the electrostatic force between two charged objects to their charges and the distance between them.

Electric Field is zero inside a conductor

: Inside a conductor, the electric field is always zero. This means that there is no net force acting on charges inside a conductor, and they are in electrostatic equilibrium.

Electric potential difference V_X - V_Y

: The electric potential difference between two points, V_X and V_Y, is the change in electric potential energy per unit charge as a charge moves from point X to point Y.

Electric Potentials

: Electric potentials refer to the amount of electrical potential energy per unit charge at any given point in an electric field. It represents how much work is required to move a positive test charge from infinity to that point without acceleration.

Gauss' Law

: Gauss' Law relates the electric flux through a closed surface to the total charge enclosed by that surface. It provides a convenient way to calculate electric fields for symmetric charge distributions.

Line of Charge

: A line of charge refers to a distribution of electric charge that is concentrated along an infinitely long, straight line.

Potential

: The potential refers to the electric potential energy per unit charge at a specific point in an electric field. It represents the work done to move a positive test charge from infinity to that point.

Potential Difference

: Potential difference, also known as voltage, is the difference in electric potential energy per unit charge between two points in an electric circuit. It represents how much work is done on each unit of charge when it moves from one point to another.

radius vector

: In physics, radius vector (r) refers to a vector that represents both magnitude and direction from an origin point to another point in space.

Ring of Charge

: A ring of charge refers to a circular loop or disc with uniformly distributed charge along its circumference.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.