Question

Consider two concentric cylindrical shells, one of radius R1, and the other of radius R2 > R1. The length of the shells is L, such that L >>> R1, R2 so we can assume that E = E_r(r) (cylindrical symmetry, or in other words, when we are between Rl and R2, the cylinder seems infinite). Assume the inner shell has a total charge -Q, the outer shell total charge +Q.  

a) Find E(r) using Gauss's law. Use a Gaussian surface that is a cylindrical shell of radius r,  Rl < r < R2 and length l. As a hint, Q enclosed inside this Gaussian surface is  

b) Find V between the plates, V= V₊ - V_- by applying the formula

Answer 1

Using Guass Law,

E A = qin / e0

A = (2pi r l )

qin = -Q x l / L

E ( 2pi r l ) = ( -Q x I / L) / e0

E = - Q / (2 pi e0 r L )

for R1 < r < R2

b)

V = (Q / 2pi e0 L) [ ln r ] r is from R1 to R2

V = (Q / 2pi e0 L) ( lnR2 - ln R1 )

V = (Q ln(R2/R1) / 2 pi e0 L )