A ground connection is made by burying a hemispherical condu

A ground connection is made by burying a hemispherical conductor of radius 25 (mm) in the earth with its base up, as shown in below figure. Assuming the earth conductivity to be 10^-6 S/m, find the resistance of the conductor to far-away points in the ground.

Solution

Current Density J = I/S = I/(2*pi*R^2)

Electric field = E = J/sigma = I/(2*pi*sigma*R^2)

potential drop V = - integral(E*dx)

V = - integral(I/(2*pi*sigma*R^2)*dR) from (infinite to a)

V = I/(2*pi*sigma*a)

by ohm\'s law

R = V/I = 1/(2*pi*sigma*a)

R = 1/(2*3.14*10^-6*25*10^-3)

R = 6.37*10^6 ohm