The electrostatic potential inside a charged spherical ball is gi… - Vidyadip

MCQ

The electrostatic potential inside a charged spherical ball is given by : $V = b -ar^2$, where $r$ is the distance from the centre ; $a$ and $b$ are constants. Then, the charge density inside the ball is :

A
$24\pi \,a{\varepsilon _0}r$
B
$6\,a{\varepsilon _0}r$
C
$24\pi \,a{\varepsilon _0}$
✓
$6\,a{\varepsilon _0}$

✓

Answer

Correct option: D.

$6\,a{\varepsilon _0}$

d
Electric filed, $E=-\frac{d \phi}{d r}=-2 a r$

By Gauss's law, $E .4 \pi r^{2}=\frac{q_{i n}}{\in_{0}}$

$\Rightarrow q_{i n}=(-2 a r) 4 \pi r^{2} \in_{0}=-8 \pi \in_{0} a r^{3}$

Now $\frac{d q_{i n}}{d r}=-24 \pi \in_{0} a r^{2}$ and $V=\frac{4}{3} \pi r^{3}, \frac{d V}{d r}=4 \pi r^{2}$

Charge density, $\rho=\frac{d q_{i n}}{d V}=\frac{d q_{i n}}{d r} \times \frac{d r}{d V}=\left(-24 \pi \in_{0} a r^{2}\right) \times \frac{1}{4 \pi r^{2}}=-6 \in_{0} a$

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