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A hollow dielectric sphere of inner radius a and outer radius 2a is uniformly charged. The relative permittivity of the material is $\in_r$ and the total charge is Q. Find The energy stored in the dielectric sphere.

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Potential at a point at a distance $r (<2a)$ from the centre due to charge enclosed by the sphere of radius of r is
$V= \large\frac{K \sigma (4/3) \pi (r^2-a^3)}{r}$
$\sigma= \large\frac{Q}{(4/3)(7 \pi a^3)} $ and
$k= \large\frac{1}{4 \pi \in_0}$
Work done in bringing an elementary shell of radius r and thickness dr is
$dW= V \sigma 4 \pi r^2 dr$
$\quad= \large\frac{K \sigma 4 \pi (r^3-a^3)}{3r} $$. \sigma 4 \pi r^2 dr$
$\qquad= \large\frac{k \sigma ^2 (4 \pi )^2}{3} $$(r^4-a^3r)dr$
=> $W= \large\frac{141 Q^2}{1960 \pi n\in_0 a}$
answered Jun 11, 2014 by meena.p

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