Question

An insulating sphere of radius a has charge density ρ(r) = ρ₀r², where ρ₀ is a constant with appropriate units. The total charge on the sphere is -3q. Concentric with the insulating sphere is a conducting spherical shell with inner radius b > a and outer radius  The total charge on the shell is +2q. Determine

(a) The magnitude of electric field at the following locations:

(i) r < a; ii) a < r < b; (iii) b < r < c; (iv) r > c.

(b) The total charge on the inner and outer surface of the shell.

(c) The surface charge density on the inner and outer surface of the shell.

(d) Extra credit: ρ₀ in terms of q, a, and the fundamental constants.

Answer 1

total charge on sphere = -3q

charge on conducting shell =2q

charge density

(a) we will use here gauss law and imagine gaussian surface

(i) for r< a

(ii) for a<r<b

(iii) for b<r<c

insider the conductor E is alway zero

so E = 0

(iv) for r>c

total charge on the surace of the system (i.e surface on the conductor shell (q') = -3q+2q )

(b) (i) -3q charge on sphere induced to on inner surface of the conductor shell

so due to induction

charge on inner conductor shell = +3q

(ii) now the charge on inner conductor shell +3q will induced on the surface of the outer conductor shell and the total charge on the outer surface of the conductor shell will be

-3q+ 2q = -q

(c) suface charge density

i) inner surface of conductor shell

(ii) outer surface of conductor shell

(d)    we  know that

for sphere