# Electric Potential around two charged hollow cylinders

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.

• The net charge on the arrangement must be zero if the there is no potential gradient when $r>d$. Commented Jul 21, 2022 at 7:19
• @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). Commented Jul 21, 2022 at 8:46
• 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 $r>d$. Commented Jul 21, 2022 at 9:01
• I'm sorry, You're right. I stated the problem vaguely. I edited it and hope it's clearer now. Commented Jul 21, 2022 at 12:06