The
voltage change from a single current electrode to some point in the half space
representing the earth is given by the expression to the right. In this expression,
V is voltage, I is current, rho is resistivity, and r
is the distance between the current electrode and the point the voltage is measured.
Notice that this expression is nothing more than Ohm's law
with the resistance, R equal to rho over 2 ¹ r.
If the Earth had a constant resistivity (it doesn't) we could estimate this resistivity by performing the following experiment. Attach to the wire connecting the battery with one of the current electrodes an ammeter to measure the amount of current going into the earth. Place one electrode connected to a voltmeter next to the current electrode and place the other at some distance, r, away from the electrode and measure the voltage difference between the two locations. Using the expression given above, compute the resistivity, rho.
In practice, this experiment could be difficult to implement because the two current electrodes must be placed a great (usually 10 times the distance over which the voltage is being measured) distance from one another. So, why not simply decrease the distance between the two voltage electrodes so the distance between the two current electrodes remains at a practical distance? The problem is that the closer the two voltage electrodes are to each other, the smaller the voltage difference that must be measured. Thus, there is a practical limit to how close the two voltage electrodes can be and thereby how close the two current electrodes can be. More importantly, there is no need to place the second electrode at "infinity" if we develop a method to take its effect into account.
As another note, one may ask why don't we simply attach the voltmeter to the wire in which the current is flowing and measure the voltage drop between the two current electrodes. This could be done. In practice, however, it is impossible to obtain information about the Earth, because what you measure is more a function of the contact resistance between the earth and the current electrodes than of the resistivity of the Earth. The contact resistance is the resistance that is encountered to current flow because the electrode does not make perfect electrical contact with the earth. Contact resistances can be quite large, on the order of kilo-ohms (10^4 ohms), although good field practice can reduce these to the order of an ohm, if necessary. However, the resistance between the electrodes will be made up of the contact resistance at each electrode, and the resistance of the Earth - which is effectively zero (the "wire" had a very large cross-sectional area!). So, the voltage measured will be dominated by the voltage drop over the electrode/earth contacts, even if it is small.
If a large (infinite) impedance voltmeter is used to make the voltage measurement across two separate voltage electrodes, however, very little current actually flows through the voltage electrodes and contact resistance is unimportant to the measurement.
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