A spherical capacitor centered on the origin is initially un

A spherical capacitor (centered on the origin) is initially uncharged. It is connected to a battery (right) which pulls charge from one conductor to the other, such that a total charge q flows onto the inner surface. What is the surface charge density, sigma_b, on the inner sphere? What is the electric field in the space between the two conductors? What is the potential in that region? What is the potential difference between the two conductors? What is the capacitance? What is the electric field outside the capacitor?

Solution

a.

total charge on inner surface = q

surface area = 4 pi b^2

surface charge density = q /( 4 pi b^2 )

b. Using Gauss Law

flux = E . A = Qinside / e0

E ( 4 pi r^2) = q / e0

E = q / (4 pi e0 r^2)

c.

V(r) = - integral of E(r) dr

V(r) = - integral of q / (4 pi e0 r^2) dr

r is from b to r.

V(r) = ( q / 4 pi e0 ) [ 1/r - 1/b ]

d.

taking V(b) = 0

V = V+ - V- = V(b) - V(a)

= (q / 4 pi e0 ) ( a - b / ab )

e.

Q = CV

C = q / ( (q / 4 pi e0 ) ( a - b / ab ) )

C = (4 pi e0 ab ) / (a - b)

f. from Gauss Law,

E . A = Qinside / e0

Q = q-q = 0

E = 0