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 [ $δ = 10^{- 3} π, I = 1 A$ ](up to two decimal places)

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Solution

$j = σ E$
Assuming that the current density is uniform in all directions ,
$\Rightarrow \frac{i}{2 π r^{2}} = σ E$
where, $2 π r^{2}$ is the surface area of the hemisphere
$\Rightarrow E = \frac{i}{2 π r^{2} σ} \Rightarrow Δ V = \int E . d r = \int_{2}^{3} \frac{i d r}{2 π r^{2} δ}$
$Δ V = \frac{i}{2 π δ} (- \frac{1}{r})_{2}^{3} \Rightarrow Δ V = \frac{1}{2 \times π \times π \times 10^{- 3}} \times \frac{1}{6} = \frac{10^{3}}{12 π^{2}}$

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