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Conductivity of Earth:Two hemispherical electrodes in earth.

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1. Two solid hemispherical conducting electrodes each of radius a are pressed into the earth, curved surfaces down, such that the flat surfaces are flush with the flat surface of the earth. Assume that the electrode separation (center to center) is d where d>>a. We wish to measure the electrical conductivity of the ground from a measurement of the current I flowing between electrodes. The two electrodes are connected above ground by a battery and a series ammeter, which completes the circuit through the earth. Assuming the earth is a homogeneous conductor with conductivity σ, find an expression for σ in terms of the measured current and the given geometry. [Hints: Consider first: whole spheres immersed in a conducting medium. Let the two spheres have charge Q and -Q respectively and find the potentials of each sphere, and the capacitance of the pair]

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Very good typical question in which two hemispherical electrodes are pressed into ground at some distance to measure electrical conductivity of earth.

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Let two spheres of radius a are placed well within the earth and connected to a voltage source creating a potential difference V between them. Considering the earth as a large homogeneous poorly conducting medium of conductivity s, a current will flow through it between the spheres prevailing the large space between them. The current lines are curves, which are normal to the symmetric plane bisecting the line joining the centers of the spheres. This medium is poorly conducting and hence behaving as a dielectric also, say with relative permitivity .

Let the perpendicular ...

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