Electrical conduction in solids II. Theory of temperature-dependent conductors

Abstract

The experiments in part I on the behaviour of the contact between metals when large currents pass the interface have yielded results which cannot be explained by the classical theory of constriction resistances. In an attempt to provide an account of this anomalous behaviour a new mathematical treatment of the general problem of the electrical heating of conductors has been developed. This treatment gives, under the appropriate conditions, a concise derivation of all the main results of the accepted theory; in addition it leads to three new conclusions. First, it is shown that the general problem of finding the spatial distribution of the potential, the current, and the temperature within a medium whose thermal and electrical conductivities vary with temperature may be reduced to the corresponding problem with constant conductivities, and a simple numerical integration. Secondly, it is shown that the spatial distribution of the current within the conductor is independent of the thermal and electrical properties of the medium, and that it is unaltered by variation s of the total current passing. Finally, it is demonstrated that there are certain conditions under which a steady solution of the problem is impossible. For many conductors steady conditions are possible only when the current is below a certain critical value. If a current greater than this critical value is maintained the temperature will rise continuously, and eventually the process will be limited by some other phenomenon, for example, melting. The new treatment is applied to the calculation of the spatial distribution of current for a particular shape of conductor, one which is relevant to many resistance welding processes. The predictions agree accurately with experimental data obtained from the examination of a series of welds. The theory is then applied to the special case of the electrical contact between gold pieces and shown to offer an explanation of the anomalous behaviour mentioned above. Good agreement is demonstrated between the theoretical predictions and the experimental results.