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Qualitative: A Uniformly Charged Cylinder

1. The problem statement, all variables and given/known data
This is just a qualitative question that is along side my main lecture with Griffiths EM book.

Basically we have a very long cylinder with charge density sigma and radius a along the z axis. This is mostly beside the point and is just to setup the question. The question asks that we are in a situation where a student thinks there are components of the E-field in every direction (for reference we use (s, phi, z) for the coordinates) outside the cylinder. It then asks you to explain each component of why you can or can't have them in, essentially, laymen terms (which I assume to be likely symmetry arguments). The s and phi I think I can do but the z I am having trouble discerning how symmetry says we can't have a z component.

2. Relevant equations


3. The attempt at a solution
To start I know the electric field is only supposed to depend on the s component. However, I am confused about why we can't put an E-field on top of the cylinder and it thus would have components in this direction. I should mention that the cylinder is supposed to be very long which I am not sure if that means we don't care about the caps. In which case then if we put a test charge outside the cylinder then we move the cylinder up and down the test charge will not be affected because it looks exactly the same. However, lets say it is of length L how would this affect the answer exactly?
 

BvU

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In this context, 'very long' means you don't have to consider end-effects. I think you get the picture. And indeed, by the time one of the ends does need to be taken into consideration, there is a z-component of the field emerging. If you go far enough, the very long cylinder starts to look like a point charge more and more and field goes towards a point-charge field. But then the book wouldn't have mentioned 'very long'.
 
Thanks for the reply.
 

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