Problem One: Consider a conducting spherical shell with an inner radius a and ou

Question

Problem One: Consider a conducting spherical shell with an inner radius a and outer radius c. Let the space between two surfaces be filled with two different dielectric materials so that the dielectric constant is K between a and b, and k2 between b and c. The charge density of the inner shell is given by K2 0 a) Calculate the surface charge density of the outer shell in terms of and other given constants b) Find the capacitance of the spherical capacitor. c) Check what happens to the capacitance in the limit when ki and K2 go to 1. Explain your result

Explanation / Answer

given

conducting spherical shell inner radius a, outer radius c

for a < r < b, dielectric constant = k1

for b < r < c, dielectric comnstant = k2

charge density of inner shell = sigma = Q/4*pi*a^2

a. outer shell charge density = -Q/4*pi*c^2

sigma' = -sigma(a/c)^2

b. capacitance = C

now

Q = CV

at radius r

dV = kQ/a - kQ/b

hence

Q = C*dV = C*kQ(1/a - 1/b)

C = 4*pi*epsilon/(1/a - 1/b)

but for the dielectric

C1 = 4*pi*epsilon*k1/(1/a - 1/b)

C2 = 4*pi*epsilon*k2/(1/b - 1/c)

and

for capacitors added in series

C = C1C2/(C1 + C2) = 4*pi*epsilon*k1*k2/(k1(1/b - 1/c) + k2(1/a - 1/b))

C = 4*pi*epsilon*k1*k2/(k1(1/b - 1/c) + k2(1/a - 1/b))

c. k1 = k2 = 1

C = 4*pi*epsilon/(1/a - 1/c)

trhe capacitor becomes a normal concentric spherical capacitor with no dielectirc

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