Question

A hemispherical earth plate is buried into the earth. Find the potential difference between the feet of a man standing on ground as shown in figure. Current through earth plate is I. Conductivity of earth is [$\delta ={10}^{-3}\pi ,I=1A$](up to two decimal places)

Answer 1

$j=\sigma E$
Assuming that the current density is uniform in all directions ,
$\Rightarrow \frac{i}{2\pi r^2}=\sigma E$
where, $2\pi r^2$ is the surface area of the hemisphere
$\Rightarrow E=\frac{i}{2\pi r^2\sigma }\Rightarrow \Delta V=\int E.dr={\int }_2^3\frac{idr}{2\pi r^2\delta }$
$\Delta V=\frac{i}{2\pi \delta }(-\frac{1}{r}{)}_2^3\Rightarrow \Delta V=\frac{1}{2\times \pi \times \pi \times {10}^{-3}}\times \frac{1}{6}=\frac{{10}^3}{12{\pi }^2}$