Question

Consider two conducting spheres of radii $R_{1}$ and $R_{2}$ with$R_{1} >\,R_{2}$. If the two are at the same potential, and the larger sphere has more charge than the smaller sphere, then

• A. the charge density of smaller sphere is less than that of larger sphere
• B. the charge density of smaller sphere is more than that of larger sphere
• C. both spheres may have same charge density
• D. none of these

Answer 1

Answer: (B)

Here, $V_{1}=V_{2}$ or $\frac{q_{1}}{4\pi\varepsilon_{0} R_{1}}=\frac{q_{2}}{4\pi\varepsilon_{0} R_{2}}$
$\therefore \frac{q_{1}}{q_{2}}=\frac{R_{1}}{R_{2}}$
Given $R_{1}>\,R_{2}$
$\therefore q_{1}>\,q_{2}$
$\therefore $ Larger sphere has more charge than the smaller sphere.Now charge densities
$\sigma_{1}=\frac{q_{1}}{4\pi R_{1}^{2}}$and $\sigma_{2}=\frac{q_{2}}{4\pi R_{2}^{2}}$
$\therefore \frac{\sigma_{2}}{\sigma_{1}}=\frac{q_{2}}{q_{1}}\frac{R_{1}^{2}}{R_{2}^{2}}$
or $ \frac{\sigma_{2}}{\sigma_{1}}=\frac{R_{2}}{R_{1}}\frac{R_{1}^{2}}{R_{2}^{2}}=\frac{R_{1}}{R_{2}}$ (using $\left(i\right)$)
As $R_{1}>\,R_{2}$ therefore $\sigma_{2}>\,\sigma_{1}$
Charge density of smaller sphere is more than the charge density of larger sphere