Question

Consider a thin spherical shell of radius $R$ with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field and the electric potential $V(r)$ with the distance $r$ from the centre, is best represented by which graph?

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

For inside points $(r\le R)$
$E=0\Rightarrow V=$ constant $=\frac{1}{4\pi {\epsilon }_0}\frac{q}{R}$
For inside points $(r\ge R)$
$E=\frac{1}{4\pi {\epsilon }_0}.\frac{q}{r^2}$ or $E\propto \frac{1}{r^2}$
and $V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r}$ or $V\propto \frac{1}{r}$
On the surface (r = R)
$V=\frac{1}{4\pi {\epsilon }_0}\frac{q}{R}$
$\Rightarrow E=\frac{1}{4\pi {\epsilon }_0}.\frac{q}{R^2}=\frac{\sigma }{{\epsilon }_0}$
where, $\sigma =\frac{q}{4\pi R^2}=$ surface charge density corresponding to above equations the correct graphs