Question

The value of distance $r_m$, inside the ball at which electric field intensity maximum is given by-

• A. $r_m=\frac{R}{3}$
• B. $r_m=\frac{3R}{2}$
• C. $r_m=\frac{2R}{3}$
• D. $r_m=\frac{4R}{3}$

Answer 1

The correct option is C $r_m=\frac{2R}{3}$

Using Gauss's law,$\int \overrightarrow{E}.\hat{n}ds=\frac{Q_{enclosed}}{ϵ}$

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

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

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

for maximum E, $\frac{dE}{dr}=\frac{{\rho }_0}{ϵ}[\frac{1}{3}-\frac{2r}{4R}]=0\Rightarrow r=\frac{2R}{3}$

thus,at $r=r_m=\frac{2R}{3}$ , $E$ is maximum.