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Question:

A charge Q is distributed over three concentric spherical shells of radii a, b, c (a<b<c) such that their surface charge densities are equal to one another. The total potential at a point at distance r from their common centre, where r<a, would be

Q4πϵ₀(a²+b²+c²)(a³+b³+c³)

Q(a+b+c)4πϵ₀(a²+b²+c²)

Q4πϵ₀(a+b+c)

Q12πϵ₀(ab+bc+ca)abc

Solution:

Potential at point P, V = kQ/a + kQ/b + kQ/c ≈ Q/a : Q/b : Q/c since σa = σb = σc ∴ Qa = [a²/a²+b²+c²]Q Qb = [b²/a²+b²+c²]Q Qc = [c²/a²+b²+c²]Q V = Q4πε₀[(a+b+c)/a²+b²+c²] ∴ correct answer is (2).