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_A ne V_B`
B. `V_C = V_B ne V_A`
C. `V_C ne V_B ne V_A`
D. `V_C = V_A = V_B`

Answer

A

Explanation

Given, c = a + b
`V_A=(KQ_A)/a+(KQ_B)/b+(KQ_C)/c`
`=(Ksigma4pia^2)/a+(K(-sigma)xx4pib^2)/b+(Kxxsigmaxx4pic^2)/(c)`
`= sigma/(epsilon_0)(a-b+c)`
`V_B=(KQ_A)/b+(KQ_B)/b+(KQ_C)/c`
`=(Kxxsigmaxx4pia^2)/(b)+(K(-sigma)xx4pib^2)/b+(Ksigmaxx4pic^2)/c`
`=(sigma)/(epsilon_0)(a^2/b-b+c)`
`V_C=(KQ_A)/c+(KQ_B)/c+(KQ_C)/c`
`=sigma/(epsilon_0)(a^2/c-b^2/c+c)`
When c = a + b
`V_A=sigma/epsilon_0(a-b+a+b) = (2sigmaa)/epsilon_0`
`V_A=sigma/epsilon_0(a^2/b-b+a+b)=sigma/epsilon_0[(a^2-ab)/b]`
`V_C=sigma/epsilon_0(a^2/c-b^2/c+a+b)=sigma/epsilon_0[((a+b)(a-b))/c +a+b]`
`=(2sigma a)/(epsilon_0)`
`V_A =V_C ne V_B`