This is a cross section of two nested cylinders. They are infinite in length goi

Question

This is a cross section of two nested cylinders. They are infinite in length going into and out of the page. The outer cylinder is a thin conductor of radius M with a uniform charge distribution and total charge -2Q. The inner cylinder is a solid insulator of radius J. It has a uniform volume charge density of and a total charge of +2Q There is only vacuum between them Region C +2Q,+p Region B 1.) (2pts) Determine the net charge on the inner and outer surfaces of the thin shell -2Q Region A 2.) (3pts) Determine the electric field in all three regions of space as a function of r, the distance from the mutual center. Use only , r, 6, and J in your expressions. You may use any or all of these as needed. 2.) (3pts) Calculate the potential change when moving from J to M.

Explanation / Answer

let length be L.

Q1.

net charge on inner surface of thin shell=-(total charge on the inner cylinder)

=-2*Q

net charge on outer surface of thin shell=2*Q-2*Q=0

Q2.

for r<J:

charge enclosed in a sphere of radius r=rho*(4/3)*pi*r^3

if electric field is E,

then using Gauss’ law:

epsilon*E*2*pi*r*L=rho*pi*r^2*L

==>E=rho*r/(2*epsilon)

for J<r<M:

charge enclosed =2*Q

using Gauss law:

epsilon*E*2*pi*r*L=2*Q

==>E=Q/(pi*epsilon*r*L)

as 2*Q=rho*total volume=rho*pi*J^2*L

E=0.5*rho*pi*J^2*L/(pi*epsilon*r*L)

=0.5*rho*J^2/(epsilon*r)

for r>M:

charge enclosed=0

electric field=0

Q3.

while moving from J to M, potential change=-integration of E*dr

with r varying from J to M

=-integration of 0.5*rho*J^2/(epsilon*r)

=(-0.5*rho*J^2/(epsilon))*ln(r)

then using the limits,

answer=0.5*rho*J^2*ln(J/M)/epsilon

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