Solution

Statement

$8.2.17.$ A metal ball of radius $r$ is located in a medium with a small conductivity $\lambda$. Determine the current flowing from the ball if its potential is $V$. If such a ball is connected by an insulated wire to a lightning rod, what will be the grounding resistance?

Solution

Current will flow from the metal ball due to charge leakage into the conducting medium. The current density on the surface of the ball is:

$$j = \lambda E$$

where $E$ is the electric field strength on the surface of the ball.

The potential of the ball is related to the electric field strength on its surface:

$$E = \frac{V}{r}$$

Then:

$$j = \lambda \frac{V}{r}$$

The current flowing from the surface of the ball is:

$$I = j \cdot S = \lambda \frac{V}{r} \cdot 4\pi r^2 = 4\pi\lambda rV$$

If the ball is connected by an insulated wire to a lightning rod (which has the potential of the Earth), then the grounding resistance is:

$$R = \frac{V}{I} = \frac{V}{4\pi\lambda rV} = \frac{1}{4\pi\lambda r}$$