Underwater Cable Resistance Problem · CAP 2010 #P2
Physics · electromagnetism
Problem
Oceanographers have deployed an array of instruments off the west coast of Vancouver Island, connected together by cables carrying power and communication. The power is supplied through a single DC line. The return path is through the sea water. The grounds at each end of the line are large metal spheres.
Consider two copper spheres, radii a, buried in an infinite medium of resistivity ρ. Current flows out of one sphere and is collected by the second sphere. The current flow in vicinity of each sphere can be considered radially uniform as the distance between the spheres is large compared with a.
a) By considering the series resistance of many thin spherical shells around a sphere show graphically (or by direct integration) that much of the resistance between the spheres is due to material within a radius 2a of each sphere and thereby estimate the total resistance between the spheres.
b) If the current flow between the spheres is 1 A, calculate the charge in coulombs on each one.