Question

Consider a spherical Gaussian surface of radius r < a. If the net charge Q(r) enclosed by this surface as a function of r is $Qr=\frac{YQ{r}^3}{a^3}$. Note that the charge inside this surface is less than $3Q$. Find out the value of $Y$?

Answer 1

Here the net charge of solid sphere of radius a is $3Q$. So the charge density will be $\rho =\frac{3Q}{\left(4/3\right)\pi a^3}$

Thus, net charge enclosed by surface of radius r where $r<a$ is

$Q\left(r\right)=\rho \frac{4}{3}\pi r^3=\frac{3Q}{\frac{4}{3}\pi a^3}\times \frac{4}{3}\pi r^3=\frac{3Q{r}^3}{a^3}$