Question

Let $P\left(r\right)=\frac{Q}{\pi R^4}r$ be the charge density distribution for a solid sphere of radius $R$ and total charge Q. For a point $p_1$ inside the sphere at distance $r_1$ from the centre of sphere, the magnitude of electric field is

• A. $0$
• B. $\frac{Q}{4\pi {ϵ}_0{r}_1^2}$
• C. $\frac{Q{r}_1^2}{4\pi {ϵ}_0{R}^4}$
• D. $\frac{Q{r}_1^2}{3\pi {ϵ}_0{R}^4}$

Answer 1

Answer: (C)

$\frac{Q{r}_1^2}{4\pi {ϵ}_0{R}^4}$

Consider a differential thickness dr at a radius r.

We get the area for this differential thickness as $dA=4\pi r^{2}dr$

Thus we get the electric field at this point as $dE=\frac{kdQ}{r_1^2}$

or

$dE=\frac{1}{4\pi {ϵ}_o}\frac{Qr4\pi r^{2}dr}{\pi R^4{r}_1^2}$

$E=\frac{Q}{4\pi {ϵ}_0{r}_1^2\pi R^4}{\int }_{r=0}^{r_1}4\pi r^{3}dr$

$=\frac{Q{r}_1^2}{4\pi {ϵ}_0{R}^4}$