Question

A non-conducting sphere of radius $R$ has a positive charge $Q$ uniformly distributed over its entire volume. A smaller, concentric, spherical volume of radius $r(r<R)$ is scooped out of the sphere. The magnitude of the electric field $E$ at a point inside the body at a distance $x$ from the centre is $(r$ $<x<R)$ is $(K=1)\left(4\pi {\in }_0\right)=$ constant $)$ :

• A. $\frac{KQ}{x^2}$
• B. $\frac{KQ}{x^2}\frac{r^3}{R^3}$
• C. $\frac{KQ}{x^2}\left(\frac{x^3-r^3}{R^3}\right)$
• D. 0

Answer 1

Answer: (C)

Charge of the remaining part after scooped out of the sphere
$=Q-\frac{Q}{\frac{4}{3}\pi R^3}\times \frac{4}{3}\pi R^3$
$=Q\left(1-\frac{r3}{R^3}\right)=\frac{Q\left(R^3-r^3\right)}{R^3}$
Now charge of the part $r<\dots ..<x$
$=\frac{Q\left(1-\frac{r3}{R^3}\right)}{\frac{4}{5}\pi \left(R^3-r^3\right)}\times \frac{4}{3}\pi \left(x^2-r^3\right)=\frac{Q}{R^3}\left(x^3-r^3\right)=Q^1$
Now field at $P=K\frac{Q^1}{x^2}=\frac{K}{x^2}$
$\frac{Q}{R^3}\left(x^3-r^3\right)=\frac{KQ}{x^2}\left(\frac{x^3-r^3}{R^3}\right)$