Research Note: A simple method of image solution for a sphere of constant electrical potential in a conducting half‐space: implications for the applied potential method

Abstract

The applied potential, or mise‐à‐la‐masse, method is used in mineral exploration and environmental applications to constrain the shape and extent of conductive anomalies. However, few simple calculations exist to help gain understanding and intuition regarding the pattern of measured electrical potential at the ground surface. While it makes intuitive sense that the conductor must come close to the ground surface in order for the lateral extent of the potential anomaly to be affected by the dimensions of the conductor rather than simply by the depth, no simple calculation exists to quantify this effect. In this contribution, a simple method of images solution for the case of a sphere of constant electrical potential in a conducting half‐space is presented. The solution consists of an infinite series where the first term is the same as the method of images solution for a point current source in an infinite half‐space. The higher order terms result from the interaction of the constant potential sphere with the no‐flux boundary condition representing the ground surface and cause the change in the shape of the potential anomaly that is of interest in the applied potential method. The calculation is relevant to applied potentials when the conductive anomaly is limited in all three space dimensions and is highly conductive. Using the derived formula, it is shown that, while the electrical potential at the ground surface caused by the sphere is affected even when the sphere is quite deep, the ratio of the potential to the current, a quantity that is more relevant to the applied potential method, is not affected until the centre of the sphere is within two radii of the ground surface. An expression for the contact resistance of the sphere as a function of depth is also given, and the contact resistance is shown to increase by roughly 45% as the sphere is moved from great depth to the ground surface.

On the determinability of all of the self‐ and mutual resistances in a grounded electrode array

“…It is also important to determine self‐resistances for grounding applications in electrical power distribution (Tagg ). Also, the size of a target can be constrained from measurements of the self‐resistance in mise‐a‐la masse surveys (Bhattacharya and Shalivahan ; Butler ). In a conventional resistivity survey, if the self‐resistances of the current electrodes were known, it might be possible to extract information from current conducting pairs and so knowledge of the self‐resistance would be useful.…”