A charges Q is distributed uniformly within the material of a hollow sphere of inner and outer radii $r_{1}$ and $r_{2}$ . Find the electric field at a point $p$ a distance $x$ away from the centre for $r_{1} < x < r_{2}$ . Draw a rough graph showing the electric field as a function of $x$ for $0 < x < 2 r_{2}$ .

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Solution

The charge density for the spherical shell is given as:
$ζ = \frac{Q}{V}$

Now, volume charge density $= \frac{Q}{\frac{4}{3} \times π \times (r_{2}^{3} - r_{1}^{3})}$
$∴ ζ = \frac{3 Q}{4 π (r_{2}^{3} - r_{1}^{3})}$

Again volume of sphere having radius $x$ is given as $= \frac{4}{3} π x^{3}$

Now charge eneclosed by the sphere having radius $x$ is:
$Q = (\frac{4}{3} π χ^{3} - \frac{4}{3} π r_{1}^{3}) \times \frac{Q}{\frac{4}{3} π r_{2}^{3} - \frac{4}{3}} π r_{1}^{3}$

$= Q (\frac{χ^{3} - r_{1}^{3}}{r_{2}^{3} - r_{1}^{3}})$

Applying Gauss;s law $- E \times 4 π x^{2} = \frac{q e n c l o s e d}{\in_{0}}$

$\Rightarrow E = \frac{Q}{\in_{0}} (\frac{χ^{3} - r_{1}^{3}}{r_{2}^{3} - r_{1}^{3}}) \times \frac{1}{4 π χ^{2}} = \frac{Q}{4 π \in_{0} χ^{2}} (\frac{χ^{3} - r_{1}^{3}}{r_{2}^{3} - r_{1}^{3}})$

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A charge Q is distributed uniformly within the material of a hollow sphere of inner and outer radii $r_{1}$ and $r_{2}$ (figure 30-E4). Find the electric field at a point P a distance x away from the centre for $r_{1} < x < r_{2}$ . Draw a rough showing the electric field as a function of x for $0 < x < 2 r_{2}$ (figure 30-E4).