Question

A spherical insulator of radius $R$ is charged uniformly with a charge $Q$ throughout its volume and contains a point charge $\frac{Q}{16}$ located at its centre. Which of the following graphs best represents qualitatively, the variation of electric field intensity $E$ with distance $r$ from the centre?

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

For $\left(x<R\right);E=\frac{Qx}{\frac{4}{3}\pi R^{3}3{\epsilon }_0}+\frac{Q/16}{4\pi {\epsilon }_0{x}^2}$
$\frac{dE}{dx}=0\Rightarrow \frac{6Q}{4\pi {\epsilon }_0}\left(\frac{1}{R^3}-\frac{2}{16x^3}\right)=0$
$\frac{d^{2}E}{d{x}^2}=\frac{6Q}{4\pi {\epsilon }_0\left(16x^4\right)}>0$
$\therefore$ At $x=\frac{R}{2}$, $E$ is minimum.
For $\left(x>R\right),E=\frac{\left(Q+Q/16\right)}{4\pi {\epsilon }_0{x}^2}$