Question

Consider a uniform charge distribution with charge density $2 \,C / m ^{3}$ throughout in space. If a Gaussian sphere has a variable radius which changes at the rate of $2 \,m / s$, then value of rate of charge of flux is proportional to $r^{k}$, ( $r=$ radius of sphere). Then, the value of $k$ is________

Answer 1

Flux linked with a Gaussian sphere of radius $r$
$=$ change enclosed $\times \frac{1}{\varepsilon_{0}}$
$=$ volume enclosed $\times$ charge density $\times \frac{1}{\varepsilon_{0}}=\frac{4}{3} \pi r^{3} \times \rho \times \frac{1}{\varepsilon_{0}}$
So, rate of change of flux $=\frac{d \phi_{E}}{d t}=\frac{d}{d t} \frac{4}{3} \pi \frac{\rho}{\varepsilon_{0}} \times r^{3}$
$=\frac{4}{3} \pi \frac{\rho}{\varepsilon_{0}} \times 3 r^{2} \frac{d r}{d t}=4 \pi r^{2} \frac{\rho}{\varepsilon_{0}}\left(\frac{d r}{d t}\right)=8 \pi r^{2} \frac{\rho}{\varepsilon_{0}}$
$\therefore \frac{d \phi_{E}}{d t} \propto r^{2}, k=2$