A solid insulating cylinder of radius A is surrounded by a hollow, concentric co

Question

A solid insulating cylinder of radius A is surrounded by a hollow, concentric conducting cylinder of inner radius B and outer radius C. The solid cylinder has a non-uniform charge density given by (r)- The conducting shell is uncharged. +pr where and are constants. a. b. c. d. What is the total charge on the solid cylinder? What is the charge on the inner surface of the conducting cylinder? What is the charge on the outer surface of the conducting cylinder? What is the electric field everywhere (r

Explanation / Answer

Given radius of insulating cylinder = A

inner radius of hollow conducting shell = B, B > A

outer radius of hollow conducting shell = C, C > B

charge density on solid cylinder, rho(r) = alpha/r + beta*r

a. consider a cylindrical shell of thickness dr at radius r < A

then

charge in this cylinder

dq = rho(r)2*pi*r*dr*l [ where l is length of the cylinder]

so Q = integrate dq fropm r = 0 to r = A

integrate(2*pi*l[alpha + beta*r^2]dr) = 2*pi*l[alpha*A + beta*A^3/3] = Q

b. as the conduicting shell cannot have electric field inside it

so if we consider a gaussean cylindrical surface with r > B, r < C, net charge enclosed should be 0

hence charge on inner surface of the cylindrical shell = -Q = -2*pi*l[alpha*A + beta*A^3/3]

c. as the outer shell was electrically neutral

charge on outer surface = 2*pi*l[alpha*A + beta*A^3/3]

d. 1. r < A

from gauss law

E*2*pi*r*l = 2*pi*l[alpha*r + beta*r^3/3]/epsilon

E = [alpha + beta*r^2/3]/epsilon

2.A < r < B

from gauss law

E*2*pi*r*l = 2*pi*l[alpha*A + beta*A^3/3]/epsilon

E = [alpha*A + beta*A^3/3]/epsilon*r

3. B < r < C

E = 0 ( electric field inside conductors is 0)

4. r > C

from gauss law

E*2*pi*r*l = 2*pi*l[alpha*A + beta*A^3/3]/epsilon

E = [alpha*A + beta*A^3/3]/epsilon*r

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