First slide

Electrostatic potential

Question

Two concentric spheres of radii R and r have similar charges with equal surface densities ( $σ$ ). What is the electric potential at their common centre

Moderate

A

$\frac{σ}{ε_{0}}$
B

$\frac{σ}{ε_{0}} (R - r)$
C

$\frac{σ}{ε_{0}} (R + r)$
D

none of the above

Solution

Let Q and q be the charges on the spheres. The potential at the common centre will be
$\begin{aligned} V = \frac{1}{4 {πε}_{0}} \times \frac{Q}{R} + \frac{1}{4 {πε}_{0}} \times \frac{q}{r} \\ = \frac{1}{ε_{0}} [\frac{Q}{4 {πR}^{2}} \times R + \frac{q}{4 {πr}^{2}} \times r] \end{aligned}$
But $\frac{Q}{4 {πR}^{2}} = \frac{q}{4 {πr}^{2}} = σ$
$∴ V = \frac{1}{ε_{0}} [σR + σr] = \frac{σ}{ε_{0}} (R + r)$

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