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This was a problem in our physics final exam and I still haven't figured it out completely. A long non-conductive cylindrical shell with inner radius "a" and outer radius "b" is surrounded by another cylindrical shell with inner radius "c" and outer radius "d" such that a<b<c<d. The inner shell has a charge density of $$\rho=-Br $$ and the outer shell has a density of $$\rho=Ar.$$ $$(A, B>0)$$

Consider infinity to have zero potential.

find the Electric potential where:

a) r>d

b) c<r<d

c) a<r<b

d) r<a

e) find the ratio of A to B such that Electric Potential would be zero in the region where r>d.

for part a, I tried to solve it in this way(I don't know if it s correct). We can find the electric field in this region using gauss's law: $$E_5=\frac {1}{3\epsilon_0 r}A(d^3-c^3)-B(d^3-a^3)$$ and then considering P as a point located in this region(r>d) with distance r from the center: $$V_p-V_\infty = +\int_{\infty}^r E_5.dr $$ And I get stuck in this part, where the right-hand integral is undefined. I don't know whether I'm making a mistake somewhere or there's something wrong with the question.

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    $\begingroup$ The net charge on the arrangement must be zero if the there is no potential gradient when $r>d$. $\endgroup$
    – Farcher
    Commented Jul 21, 2022 at 7:19
  • $\begingroup$ @Farcher That's true but the question required us to derive an expression for the electric potential for region r>d and also inner regions(c<r<d and etc). $\endgroup$
    – Sadegh Rizi
    Commented Jul 21, 2022 at 8:46
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    $\begingroup$ Considering the infinity to have zero potential, find the ratio of A to B such that Electric Potential is zero in the region where r>d. In your post you said that the potential is zero when $r>d$. $\endgroup$
    – Farcher
    Commented Jul 21, 2022 at 9:01
  • $\begingroup$ I'm sorry, You're right. I stated the problem vaguely. I edited it and hope it's clearer now. $\endgroup$
    – Sadegh Rizi
    Commented Jul 21, 2022 at 12:06

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For infinite cylinder symmetry taking the reference of potential at infinite distance is known to produce this "effect". Same as for infinite plane of charge or infinite line of charge. The effect of the charges does not go to zero fast enough as the distance goes to infinity due to the infinite character of the distribution itself. As the distribution is inifnite, there is charge at the "inifinity" so the potential cannot be zero there. Of course, this is not a problem in reality because you won't really see infinite distributions of charges. Here is an example of the way to deal with the potential of an infinite line of charge. The idea is that you take the reference somewhere else. Usualy on one of the cylindrical surfaces.

https://books.physics.oregonstate.edu/GSF/vlineinf.html#:~:text=The%20potential%20is%20a%20continuous,move%20away%20from%20the%20charge.

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