The Resistance of an Infinite Slab with a Disk Electrode

We consider the resistance of an infinite conducting slab as measured between an electrode entirely covering one face and a circular electrode affixed to the other face by a resistive film. This resistance can be imagined to be made up of two resistances in series: namely, the film contact resistance and a resistance which is due to the body to which the electrode is attached, but which depends on the film resistance. Lewis1 has derived general upper and lower bounds on this body resistance. In the present case the upper bound may be calculated analytically. The lower bound is the resistance which would exist between the electrodes in the absence of the film. Calculation of the last-mentioned resistance involves a classical potential problem treated by Weber in 1873, but still not completely solved today. We treat this problem by a combination of analysis and experiment, the latter in effect being an analog computation using an electrolytic tank. An asymptotic solution is found which converges rapidly for slab thicknesses as small as one disk radius, while for smaller thicknesses experimentally determined values of resistance are used. The upper and lower bounds for the body resistance, which differ only by 8 per cent for a thick slab and tend to the same value for a thin slab, 1609