A mathematical treatment of the general problem of electrical contacts and heating related to MEMS switch contacts is developed. The spatial distri bution of the potential, the current and the temperature in a special case of electrical contact between two gold bodies whose thermal and electrical conductivities vary with temperature is analyzed, and an explanation for the collapse of gold contact system before reaching the melting point is given. For this purpose three different methods are applied. First, we develop an exact solution, which involves with solving a dual integral equation leading to a Fred holm integral equation, that is solved numerically. Secondly, a simple model to obtain an approximate resistance of an a-spot system is presented. Finally, another approximate solution is developed, where the variational method is implemented to reduce the complicated computations of the exact method. Excellent agreement is demonstrated between the results of the exact and the variational methods for all sizes of a-spots in contact systems
