Three concentric spherical shells have radii `a, b` and `c(a lt b lt c)` and have surface charge densities `sigma, -sigam` and `sigma` respectively. If `V_(A), V_(B)` and `V_(C)` denote the potentials of the three shells, then for `c = q + b`, we have

Options

A. `V_(C) = V_(B) = V_(A)`
B. `V_(A) = V_(C) != V_(B)`
C. `V_(C) = V_(B) != V_(A)`
D. `V_(C) = V_(B) != V_(A)`

Answer

B

Explanation

As is clear from Fig.

`V_(A) = (sigma a)/(in_(0)) - (sigma b)/(in_(0)) + (sigma c)/(in_(0)) = (sigma)/(in_(0)) (a-b+C)`
As `c = a+b`
`,. V_(A) = (sigma)/(in_(0)) (2a)`
`V_(B) = (sigma)/(in_(0)) [(a^(2))/(b) - b + c] = (sigma)/(in_(0)) [(a^(2))/(b) - b + a + b]`
`V_(B) = (sigma)/(in_(0)) [(a^(2))/(b)+a] = (sigma a)/(in_(0)) ((a+b))/b) = (sigma_(0) ac)/(b)`
`V_(C) = (sigma)/(in_(0)) [(a^(2))/(c ) -(b^(2))/(c ) +c] = (sigma)/(in_(0)) [(a^(2) - b^(2) + c^(2))/(c )]`
`= (sigma)/(in_(0)) [((a+b)(a-b) + (a+b)^(2))/(c )]`
`V_(C) = (sigma)/(in_(0)) (a+b)/( c) (a-b+a+b) = (sigma 2a)/(in_(0))`
We find that `V_(A) = V_(C) = V_(B)`
Choice (d) is correct.