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 density6 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)

The charge density6 of smaller sphere is more than that of larger sphere.

Answer 2

: Here, $V_{1} = V_{2}$

or $\frac{q_{1}}{4\pi\varepsilon_{o}R_{1}} = \frac{q_{2}}{4\pi\varepsilon_{0}R_{2}}$

$\therefore\frac{q_{1}}{q_{2}} = \frac{R_{1}}{R_{2}}$

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 $\sigma_{2} = \frac{q_{2}}{4\pi R_{2}^{2}}$

$\therefore\frac{\sigma_{2}}{\sigma_{1}} = \frac{R_{2}}{R_{1}}\frac{{R_{1}}^{ÿ}}{{R_{2}}^{2}}$

or $\frac{\sigma_{2}}{\sigma_{1}} = \frac{R_{2}}{R_{1}}\frac{{R_{1}}^{2}}{{R_{2}}^{2}} = \frac{R_{1}}{R_{2}}$ (using (i)a)

As $R_{1} > R_{2}$ therefore $\sigma_{2} > \sigma_{1}$

Charge density of smaller sphere is more than the charge density of larger sphere.