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The electrostatic potential $V$ at a point on the circumference of a thin non-conducting disk of radius $r$ and uniform charge density $\sigma$ is given by equation $V = 4 \sigma r$. Which of the following expression correctly represents electrostatic energy stored in the electric field of a similar charged disk of radius $R$?
$U = \frac{8}{3}\pi {\sigma ^2}{R^3}$
$U = \frac{4}{3}\pi {\sigma ^2}{R^3}$
$U = \frac{2}{3}\pi {\sigma ^2}{R^3}$
None of these.
Solution
Charge in layer of width $\mathrm{dr}$
$\mathrm{dq}=2 \pi \mathrm{rdr} \sigma$
Work to bring $\mathrm{dq}=\mathrm{dU}=\mathrm{Vdq}$
$\mathrm{dU}=4 \sigma r \cdot 2 \pi \mathrm{rdr} \sigma=8 \pi \sigma^{2} \mathrm{r}^{2} \mathrm{dr}$
$\mathrm{U}=8 \pi \sigma^{2} \int_{0}^{\mathrm{R}} \mathrm{r}^{2} \mathrm{dr}=\frac{8 \pi \sigma^{2}}{3} \mathrm{R}^{3}$
