Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densities +σ, −σ and +σ respectively. The potential of shell B is :

$\frac{σ}{ε_{o}} [\frac{b^{2} - c^{2}}{c} + a]$

$\frac{σ}{ε_{0}} [\frac{a^{2} - b^{2}}{a} + c]$

$\frac{σ}{ε_{0}} [\frac{a^{2} - b^{2}}{b} + c]$

$\frac{σ}{ε_{0}} [\frac{b^{2} - c^{2}}{b} + a]$

Solution

$\frac{σ}{ε_{0}} [\frac{a^{2} - b^{2}}{b} + c]$

Outside the sphere (P lies outside the sphere)

$E_{o u t} = \frac{1}{4 π ε_{0}} \frac{Q}{r^{2}} = \frac{σ R^{2}}{ε_{0} r^{2}} V_{o u t} = \frac{1}{4 π ε_{0}} \frac{Q}{r^{2}} = \frac{σ R^{2}}{ε_{0} r}$

where,

σ-surface charge density

At the surface of Sphere,

V= R

$E_{s} = \frac{1}{4 π ε_{0}} \frac{Q}{R^{2}} = \frac{σ}{ε_{0}} V_{s} = \frac{1}{4 π ε_{0}} \frac{Q}{R} = \frac{σ R}{ε_{0}}$

$V_{B} = V_{B (I)} + V_{B (I I)} + V_{B (I I I)} V_{B} = \frac{K 4 π a^{2} σ}{b} + \frac{K 4 π b^{2} (- σ)}{b} + \frac{K 4 π c^{2} σ}{c} = \frac{σ}{ε_{0}} (\frac{a^{2} - b^{2}}{b} + c)$

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