Resistance of a Washer

A washer is made of a dielectric of resistivity ρ. It has a square cross section of length on a side, and its outer radius is 2a. A small slit is made on one side and wires of negligible resistance are connected to the faces exposed by the slit (see Figure 1.1). If the wires were connected into a circuit, what would be the lumped resistance due to the washer?

Figure 1.1

SOLUTION

From the cylindrical symmetry of the washer, there is no radial dependence of the potential. We can therefore consider infinitesimal current rings flowing through the washer and sum them to obtain the total current and thereby the lumped resistance. Given that the potential difference between the faces of the cut washer is V, we have

(1)

where r is the radius of one of the current rings. Integrating (1) from the inner to the outer radius,

so the resistance

The resistance of the washer is bounded by the resistances of a bar a × a in cross section and either 2πa or 4πa in length: