Question

Consider a uniform charge distribution with charge density $2C/m^3$ throughout in space. If a Gaussian sphere has a variable radius which changes at the rate of $2m/s$, 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

Answer: 2

Flux linked with a Gaussian sphere of radius $r$
$=$ change enclosed $\times \frac{1}{{\epsilon }_0}$
$=$ volume enclosed $\times$ charge density $\times \frac{1}{{\epsilon }_0}=\frac{4}{3}\pi r^3\times \rho \times \frac{1}{{\epsilon }_0}$
So, rate of change of flux $=\frac{d{\varphi }_E}{dt}=\frac{d}{dt}\frac{4}{3}\pi \frac{\rho }{{\epsilon }_0}\times r^3$
$=\frac{4}{3}\pi \frac{\rho }{{\epsilon }_0}\times 3r^2\frac{dr}{dt}=4\pi r^2\frac{\rho }{{\epsilon }_0}\left(\frac{dr}{dt}\right)=8\pi r^2\frac{\rho }{{\epsilon }_0}$
$\therefore \frac{d{\varphi }_E}{dt}\propto r^2,k=2$