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 $= \frac{1}{4 \pi\varepsilon_0} \frac{q}{R}$
For inside points $(r \ge R)$
$E = \frac{1}{4 \pi\varepsilon_0}.\frac{q}{r^2}$ or $E \propto \frac{1}{r^2}$
and $V = \frac{1}{4 \pi\varepsilon_0}\frac{q}{r}$ or $V \propto \frac{1}{r}$
On the surface (r = R)
$V = \frac{1}{4 \pi\varepsilon_0}\frac{q}{R}$
$\Rightarrow E = \frac{1}{4 \pi\varepsilon_0}.\frac{q}{R^2}=\frac{\sigma}{\varepsilon_0}$
where, $\sigma = \frac{q}{4 \pi R^2}=$ surface charge density corresponding to above equations the correct graphs