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Unit 1

1.5 Other Charge Distributions - Fields & Potentials

2 min readโ€ขnovember 3, 2020

peter57616

Peter Apps


1.5: Other Charge Distributions - Fields & Potentials

This topic is a catch-all for more advanced applications of Charge Distributions, Gauss' Law, and calculating Electic Potentials. 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 charge distributions are cases where the charge is in a ring, or line, or sheet.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2Fparagraph%201-aEtxGtrrrXAg.PNG?alt=media&token=02c4a030-e620-4cd8-97d8-9a8c5e708acd
Ring of Charge 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
Line of Charge 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

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-aBRsjB7CBRq8.PNG?alt=media&token=a0240dee-ec46-4e44-bf39-78be61c84aca
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-5f4VIM3lY5NL.PNG?alt=media&token=6afdf33c-fcbe-4095-b192-b0e6fec88555

Gauss' Law for Various Shapes

You should be able to use Gauss' Law to derive these if you need to. Check out hyperphysics for more info.
Line of Charge:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Beuigbr2OzdR.PNG?alt=media&token=7545bab0-b520-4248-9063-fc03ccb94a60
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6gKmJ2Sfjahw.PNG?alt=media&token=413feae2-c714-41cd-8798-6f526d7815ca
Point, Hoop, or Sphere (fully enclosed):
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-RUAfutMtZV7b.PNG?alt=media&token=c21869f5-6529-4cd5-95ac-edc50ae1fe94
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
Note for Spheres: The electric field inside a uniformly charged sphere is proportional to r until r = R, then it behaves as an inverse square.
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%2F-DJnOGUvsiQKY.PNG?alt=media&token=f31a2408-f85c-440c-bf9a-a8144095fc91

Potential Difference for a Variety of Shapes

This is more straightforward. For all the shapes, simply apply Gauss' Law 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
Line of Charge Example:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-HJ00jtXtQXEB.PNG?alt=media&token=7acbc9ae-2c58-4a68-a59f-b00be995601b
Conducting Sheets 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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-JDKv2I6ocqi7.PNG?alt=media&token=78444203-9ee4-45ad-bd77-99083c62f798
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:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-exw063w1UWuc.PNG?alt=media&token=8ccaa618-d0c4-421b-b4b6-2992d7cc582b
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 potential looks like a positively charged sphere (constant when r < a, and proportional to 1/r when r > a)

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