Question

The magnitude of electric field as a function of the distance r inside the sphere is given by :

• A. $E=\frac{{\rho }_0}{\epsilon }[\frac{r}{3}-\frac{r^2}{4R}]$
• B. $E=\frac{{\rho }_0}{\epsilon }[\frac{r}{4}-\frac{r^2}{3R}]$
• C. $E=\frac{{\rho }_0}{\epsilon }[\frac{r}{3}+\frac{r^2}{4R}]$
• D. $E=\frac{{\rho }_0}{\epsilon }[\frac{r}{4}+\frac{r^2}{3R}]$

Answer 1

The correct option is A $E=\frac{{\rho }_0}{\epsilon }[\frac{r}{3}-\frac{r^2}{4R}]$

By Gauss's law, the field at a distance r inside sphere is

$E.4\pi r^2=\frac{Q_{in}}{{ϵ}_0}$

here, $Q_{in}={\int }_0^r\rho \left(4\pi r^{2}dr\right)=4\pi {\rho }_0{\int }_0^r[1-r/R]r^{2}dr=4\pi {\rho }_0[r^3/3-r^4/4R]$

Thus, $E.4\pi r^2=\frac{4\pi {\rho }_0[r^3/3-r^4/4R]}{{ϵ}_0}$

or $E=\frac{{\rho }_0}{{ϵ}_0}[r/3-r^2/4R]$