Question

A spherical shell of radius R carries a uniformly distributed charge q. The electrical forces arising cause the expansion of the shell. The mechanical stress in the shell is

• A. $\frac{q^2}{32\pi^2\epsilon_0 R^4}$
• B. $\frac{q^2}{16\pi^2\epsilon_0 R^4}$
• C. $\frac{q^2}{4\pi^2\epsilon_0 R^4}$
• D. $\frac{q^2}{2\pi^2\epsilon_0 R^4}$

Answer 1

Answer: (A)

$\frac{q^2}{32\pi^2\epsilon_0 R^4}$

Answer 2

The pressure on the surface due to the charge is given by

$$P=\frac{\sigma^2}{2\epsilon_0}$$

where the surface charge density $\sigma=\frac{q}{4\pi R^2}$.

Substitute $\sigma$:

$$P=\frac{1}{2\epsilon_0}\left(\frac{q}{4\pi R^2}\right)^2=\frac{q^2}{2\epsilon_0 \cdot 16\pi^2 R^4}=\frac{q^2}{32\pi^2\epsilon_0 R^4}.$$

Thus, the mechanical stress in the shell (which must counterbalance this outward electrostatic pressure) is

$$\frac{q^2}{32\pi^2\epsilon_0 R^4}.$$