Question

1. A cylindrical capacitor is made of a conducting inner cylinder of radius R and a conducting shell of inner radius 2R and outer radius 3R. The space in between the inner cylinder and the shell is filled with a uniform dielectric material of dielectric constant k. The inner cylinder and the cylindrical shell carry equal and opposite charges but with charge per unit area of ? on the inner cylinder and -?′on the shell.

   1. (a) Find the electric field as a function of r in all regions (r < R, R < r < 2R, 2R < r < 3R, and r > 3R). (1.5 points)

   2. (b) What is the potential difference between the inner cylinder and the shell? (1 point)

   3. (c) What is the capacitance per unit length of the capacitor? (1 point)

   4. (d) If we replace the dielectric with a material with resistivity of ? = ?? , where a is a given constant and r is

      measured radially from the central axis, what is the resistance of length L of the material. (1.5 point)

Answer 1

given,

inner radius of conducting cylinder R
inner radius of outer shell 2R
outer radius of outer shell 3R

0 < r < R is conducting
R < r < 2R is filled with dielectric of constant k
2R < r < 3R is again conducting cylinder

now, capacitance of cylinderical capacitor is given by
C = 2*pi*epsilon0*L/ln(b/a)
where L is length of cylindrical capacitro and b and a are outer and inner radii

so assume the length of capacitor is L
and the given arrangement can be considered as two capacitors in series

so for the inner capacitor
0 < r < 2R
C1 = 2*pi*epsilon0*k*L/ln(2)

charge density on inner shell is sigma
sigma = Q/2*pi*RL

charge density on outer shell is sigma'
sigma'= Q/4*pi*RL
comparing, sigma'= sigma/2

a. for 0 < r < R
electric field is 0
because electric field inside conductors is 0
for R < r < 2R
E = Q/(r - R)C = 2*pi*R*L*sigma/(C*(r - R))
E = 2*pi*R*L*sigma*ln(2)/(2*pi*epsilon0*k*L*(r - R))
E = R*sigma*ln(2)/(epsilon0*k*(r - R))

for 2R < r < 3R, E = 0
for r > 3R , E = 0 (assuming the outer shell has charge -Q)

b. V = Q/C = sigma*R*ln(2)/epsilon0*k
c. capacitance per unit length is
C = C1/L = 2*pi*epsilon0*k/ln(2)

d. if dielectric is replaced by material of resistivity rho = ar

then
for R < r < 2R
dR = rho*dr/(2*pi*rL)
dR = a*dr/(2*pi*L)
integrating
R = a(2R - R)/(2*pi*L)
R (resitance) = aR/(2*pi*L)
resistance of length L is R = aR/(2*pi*L)
where L is length, R is radius given, a is constnat