Three concentric spherical metallic shells A, B and C of radii a, b and c (a < b < c) have surface charge densities `sigma`, `-sigma` and `sigma` respectively.
`(i) Find the potential of the three shells A, B and C.
(ii) If the shells A and C are at the same potential, obtain the relation between the radii a, b and c.

Options

A. `(a)/(2) log (1 - (Q)/(2 pi a A)))`
B. `(a)/(2) log ((1)/(1-(Q)/(2 pi a))`
C. `a log (1-(Q)/(2 pi a A))`
D. `a log ((1)/(1 - (Q)/(2 pi a A)))`

Answer

B

Explanation

Potential of B = Potential due to charge on A + Potential due to charge on B + Potential due to charge on C.
`therefore V_B=(k(Q_A+Q_B))/b+(kQ_c)/c`
`=1/(4 pi e_0) [(sigma 4 pi a^2)/b- (sigma 4 pi b^2)/b+ (sigma 4 pi c^2)/c] ,= sigma/e_0 ((a^2-b^2)/b+c^2/c) V_B=sigma/e_0 ((a^2-b^2)/b+c)`