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Three concentric metallic shells $A, B, a n d C$ of radii $a, b a n d c (a < b < c)$ have surface densities $σ, - σ a n d σ$ respectively. If the shells A and C are at the same potential then the relation between a, b and c is

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c = a + b

a = b + c

b = a + c

None

answer is A.

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Detailed Solution

$\begin{matrix} V_{A} = V_{C} \end{matrix}$

$\begin{matrix} \frac{k σ 4 π a^{2}}{a} - \frac{k σ 4 {πb}^{2}}{b} + \frac{k σ 4 {πc}^{2}}{c} = \frac{k σ 4 π a^{2}}{c} - \frac{k σ 4 {πb}^{2}}{c} + \frac{k σ 4 {πc}^{2}}{c} \end{matrix} (∵ v = \frac{k q}{r} & q = σ 4 {πr}^{2})$

$k σ 4 π (a - b + c) = kσ 4 π \frac{(a^{2} - b^{2} + c^{2})}{c}$

$(ac - bc + c^{2}) = (a^{2} - b^{2} + c^{2})$

$(a - b) c = (a - b) (a + b)$

$\Rightarrow c = a + b$

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