Question

Derive expressions for electric potential due to a uniformly charged spherical non conducting solid sphere at points (i) Outside (ii) sphere at points (iii) Inside the sphere. Draw the graph showing variation of potential with distance.

Accepted Answer

Let the charge on the sphere be $Q$ and its radius be $R$ . At point A, which is at a distance $r$ from the centre of the sphere, the Gaussian surface will enclose the entire surface of the nonconducting solid sphere.

From Gauss Law, the flux through the Gaussian surface will be : $ϕ = \frac{Q}{ε_{0}}$
$E (4 π r^{2}) = \frac{Q}{ε_{0}} \Rightarrow E = \frac{Q}{4 π ε_{0} r^{2}}$

$E = - \frac{d V}{d r} \Rightarrow V = - \int E . d r \Rightarrow V = - \int \frac{Q}{4 π ε_{0} r^{2}} d r V = - \frac{Q}{4 π ε_{0}} \int \frac{d r}{r^{2}} \Rightarrow V_{A} = \frac{Q}{4 π ε_{0} r}$

At (B) which lies on the surface of the sphere, the charge enclosed within the Gaussian surface will be the entire charge on the sphere, thus from Gauss law: $ϕ = \frac{Q}{ε_{0}}$

$E (4 π R^{2}) = \frac{Q}{ε_{0}} \Rightarrow E = \frac{Q}{4 π ε_{0} R^{2}}$

$V = - \int E . d r = - \int_{0}^{R} \frac{Q}{4 π ε_{0} R^{2}} . d r V_{B} = \frac{Q}{4 π ε_{0} R} = \frac{k Q}{R}$

At (C) which lies within the sphere at a distance $r$ , the Gaussian surface will not hold the entire charge, the charge enclosed within the Gaussian surface will be proportional to its volume. Applying the gauss law :

$ϕ = \frac{Q_{i n}}{ε_{0}} = \frac{Q}{ε_{0} \frac{4}{3} π R^{3}} \times \frac{4}{3} π r^{3} = \frac{Q r^{3}}{ε_{0} R^{3}}$
$E (4 π r^{2}) = \frac{Q r^{3}}{ε_{0} R^{3}} = \frac{Q r}{4 π ε_{0} R^{3}}$

As we go from B to C , work will be done against the change in potential.

$d W = E . d s \cos θ = E . d s \cos (π) \Rightarrow d W = - \frac{Q r}{4 π ε_{0} R^{3}} . d r \Rightarrow W = - \frac{k Q}{R^{3}} \int_{r}^{R} r . d r = - \frac{k Q}{2 R^{3}} [r^{2} - R^{2}] \Rightarrow V_{C} - V_{B} = - \frac{k Q}{2 R^{3}} [r^{2} - R^{2}] V_{C} - \frac{k Q}{R} = - \frac{k Q}{2 R^{3}} [r^{2} - R^{2}] = \frac{k Q}{2 R^{3}} [- r^{2} + 3 R^{2}]$

At the centre of the sphere : $r = 0$ , $V_{0} = \frac{3 k Q}{2 R}$
It can be represented as:

Derive expressions for electric potential due to a uniformly charged spherical non conducting solid sphere at points (i) Outside (ii) sphere at points (iii) Inside the sphere. Draw the graph showing variation of potential with distance.

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