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