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2. Electric Potential and Capacitance
medium
A charge is spread non-uniformly on the surface of a hollow sphere of radius $R$, such that the charge density is given by $\sigma=\sigma_0(1-\sin \theta)$, where $\theta$ is the usual polar angle. The potential at the centre of the sphere is
A
$\frac{Q}{2 \pi \varepsilon_0 R}$
B
$\frac{Q}{\pi \varepsilon_0 R}$
C
$\frac{Q}{8 \pi \varepsilon_0 R}$
D
$\frac{Q}{4 \pi \varepsilon_0 R}$
(KVPY-2009)
Solution
(d)
Potential due to an elemental charge $d Q$ at centre of sphere is
$d V=\frac{k d Q}{R}$
Potential due to complete charge distribution at centre is
$V=\oint d V=\oint \frac{k d Q}{R}$
$=\frac{k}{R} \oint d Q=\frac{kQ}{R}$
or
$V=\frac{k Q}{R}=\frac{Q}{4 \pi \varepsilon_0 R}$
Standard 12
Physics
