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2. Electric potential and capacitance — Physics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 SciencePhysics2. Electric potential and capacitance1 Mark
MCQ
The electrostatic potential inside a charged spherical ball is given by $\phi= a{r^2} + b$ where $r$ is the distance from the centre and $a, b$ 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 field

$E=-\frac{d \phi}{d r}=-2 a r.........(i)$

By Gauss's theorem

$E=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{r^{2}}.........(ii)$

From $(i)$ and $(ii)$,

$\mathrm{q}=-8 \pi \varepsilon_{0} a r^{3}$

$\Rightarrow \quad d q=-24 \pi \varepsilon_{0} a r^{2} d r$

Charge density, $\rho=\frac{d q}{4 \pi r^{2} d r}=-6 \varepsilon_{0} a$

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