A spherical capacitor is composed of an inner sphere which has a radius R 1 and

ID: 1658922 • Letter: A

Question

A spherical capacitor is composed of an inner sphere which has a radius R1 and a charge +Q and an outer concentric spherical thin shell which has a radius R2 and a charge Q.

(a) Find the electric field and the energy density as a function of r, where r is the distance from the center of the sphere, for 0 r . (Use the following as necessary: k, Q, r, and .)

kQr2

(b) Calculate the energy associated with the electrostatic field in a spherical shell between the conductors that has a radius r, a thickness dr, and a volume 4r2dr. (Use the following as necessary: k, Q, r, and dr.)
dU =.5kQ2r2dr

(c) Integrate your expression from Part (b) to find the total energy and compare your result with the result obtained using U = ½QV. (Use the following as necessary: Q, k, , R1, and R2.)

Er<R1 =

ur<R1 =

ER1<r<R2 =

kQr2

uR1<r<R2 = Er>R2 =

ur>R2 =

Explanation / Answer

given, radius of inner sphere = R1, charge = Q

Outer spherical thin shell radius = R2, charge = -Q

a. for r < R1

E = 0 ( electric field inside conductor is 0)

so as E = 0, so energy density u = 0

for R1 < r < R2

form gauss law

E*4*pi*r^2 = Q/epsilon ( where epsilon is permittvity of free space)

so E = Q/4*pi*r^2*epsilon

now k = 1/4*pi*epsilon = coloumbs constant

E = kQ/r^2

u = 0.5*epsilon*E^2

u = 0.5*epsilon*k^2Q^2/r^4

u = 0.5*epsilon*Q^2/16*pi^2*epsilon^2*r^4

u = Q^2/32*pi^2*epsilon*r^4

for r > R2

From gauss law

E *4*pi*r^2 = qen/epsilon = (Q - Q)/epsilon = 0

so E = 0

and u = 0

b. energy associated with electric field fopr radius r and thickness dr ( R1 < r < R2)

dE = u(r)*4*pi*r^2*dr

dE = Q^2*4*pi*r^2*dr/32*pi^2*epsilon*r^4

dE = Q^2*dr/8*pi*epsilon*r^2

c. total energy = integrate dE form R1 to R2

E = Q^2[1/R2 - 1/R1]/8*pi*epsilon

now capacitance of spherical capacitor is given by

C = 4*pi*epsilon(R1R2/(R2 - R1))

for Voltage applied V, Charge Q

Energy stored in capacitor = 0.5CV^2 = 0.5Q^2/C

E = 0.5*Q^2(1/R2 - 1/R1)/4*pi*epsilon

E = Q^2(1/R2 - 1/R1)/8*pi*epsilon

hence the two energies match

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A spherical capacitor is composed of an inner sphere which has a radius R 1 and

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