Question

Surface density of charge on a sphere of radius 'R' in terms of electric intensity 'E' at a distance 'r' in free space is (${ϵ}_0=$ permittivity of free space) :

• A. ${ϵ}_{0}E{\left(\frac{R}{r}\right)}^2$
• B. $\frac{{ϵ}_{0}ER}{r^2}$
• C. ${ϵ}_{0}E{\left(\frac{r}{R}\right)}^2$
• D. $\frac{{ϵ}_{0}Er}{R^2}$

Answer 1

The correct option is C ${ϵ}_{0}E{\left(\frac{r}{R}\right)}^2$

Electric field at a distance $r$ due to a charged sphere, $E=\frac{Q}{4\pi {ϵ}_o{r}^2}$

where $Q$ is the charge on the sphere

We get $Q=4\pi {ϵ}_o{r}^{2}E$

Surface area of the sphere $A=4\pi R^2$

Thus surface charge density on the sphere $\frac{Q}{A}=\frac{4\pi {ϵ}_o{r}^{2}E}{4\pi R^2}$

$⟹$ $\frac{Q}{A}=\frac{{ϵ}_{o}E{r}^2}{R^2}$