Solution
The electrostatic force on a part of a charged conductor can be calculated using the concept of electrostatic pressure. The pressure $P$ on the surface of a conductor with surface charge density $\sigma$ is:
$$P = \frac{\sigma^2}{2\epsilon_0}$$The net force on a spherical cap cut by a plane is given by the product of this pressure and the projected area of the cap onto that plane.
1. Surface Charge Density $\sigma$:
$$\sigma = \frac{Q}{4\pi R^2}$$2. Projected Area $A_{proj}$:
The cap is defined by a plane at distance $r$ from the center. The base of this cap is a circle of radius $\sqrt{R^2 – r^2}$. Thus, the projected area is:
3. Calculated Force:
$$F = P \times A_{proj} = \left( \frac{\sigma^2}{2\epsilon_0} \right) \pi (R^2 – r^2)$$ $$F = \frac{1}{2\epsilon_0} \left( \frac{Q}{4\pi R^2} \right)^2 \pi (R^2 – r^2)$$ $$F = \frac{1}{2\epsilon_0} \frac{Q^2}{16\pi^2 R^4} \pi (R^2 – r^2)$$ $$F = \frac{Q^2}{32\pi \epsilon_0 R^4} (R^2 – r^2)$$ $$F = \frac{Q^2}{32\pi \epsilon_0 R^2} \left( 1 – \frac{r^2}{R^2} \right)$$Answer: (d)