Question

A metallic sphere of radius $1m$ is given a negative charge of $20\times 10^3\mu C$ in air. If the bulk modulus is $\left(\frac{1}{4{\pi }^2}\right)\times 10^{11}N/m^2$, volume strain in the sphere is _______$\times 10^{-5}$

Answer 1

Answer: 5.65

Bulk Modulus $K=\frac{\text{ Volume stress }}{\text{ Volume strain }}$
Electrostatic pressure $P=\frac{{\sigma }^2}{2{\epsilon }_{0}k}$
$\therefore K=\frac{{\sigma }^2}{2k{\epsilon }_0(\text{ volume strain })}$
$\therefore$ Volume strain $=\frac{{\sigma }^2}{2k{\epsilon }_0}\times \frac{1}{K}$
$={\left(\frac{Q}{4\pi R^2}\right)}^2\times \frac{1}{2k{\epsilon }_0}\times \frac{1}{K}$
$\dots .\left(\because \sigma =\frac{Q}{A}\right)$
$=\frac{Q^2}{16{\pi }^2\times R^4\times 2\times {\epsilon }_0\times \frac{1}{4{\pi }^2}\times 10^{11}}$
$\dots .($ For air $k=1)$
$=\frac{{\left(20\times 10^{-3}\right)}^2}{8\times 1^4\times 8.85\times 10^{-12}\times 10^{11}}$
$=\frac{400\times 10^{-6}}{8\times 8.85\times 10^{-1}}$
$=\frac{50}{8.85}\times 10^{-5}$
$=5.649\times 10^{-5}$
$\therefore \text{ Volume strain }=5.65\times 10^{-5}$
....(Rounding off to $2$ decimal places)