Three concentric conducting shells of radii a, b and c are shown in (Fig. 3.100). Charge on the shell of radius b is Q. If the key K is closed, find the charges on the innermost and outermost shells and the radio of charge densities of the shells. Given that `a : b : c = 1 : 2 : 3`.

Explanation

After closing the key, the innermost and outermost shells will be at the same potential. Let the charge on the outer shell be q and that on the inner shell be (-q), then the total charge on inner and outer shells is zero.

potential on the innermost shell
`V_a` = Sum of potentials due to -q, Q and q
=`-q/a + Q/b + q/c`
Similarly, the potential on the outermost shell
`V_c = q/c + Q/( c) + q/( c)`
As `V_a = V_c`, we have
`-q/a + Q/b + q/( c) = -q/( c) + Q/( c) + q/c`
From the given conditions, `c = 3a, b = 2a`.
Equation (i) now becomes
`-q/a + Q/(2a) = -q/(3 a) + Q/(3 a)`
or `q = Q/(4)`
Thus, the charge on the outermost shell `= Q//4`, and the charge on the innermost shell is `-(Q//4)`.
`sigma_a = (1)/(4 pi a^2) (-( Q)/(4))`
`sigma_b = (+ Q)/(4 pi b^2) = Q/(4 pi (4 a^2))`
`sigma_c = (1)/(4 pi c^2) (Q/(4)) = (+Q)/(16 pi (9 a^2))`.