A higher dimensional theory of electrical contact resistance

The electrical contact resistance is computed for a local constriction of finite length and finite transverse dimension in a conducting current channel. Conformal mapping is used for a rectangular current channel, and an electrostatic code is used for a cylindrical current channel. The connecting bridge, which models a local electrical contact, is assumed to be made of the same conducting material as the main current channel. Very simple analytic scaling laws for the contact resistance are constructed for a wide range of geometrical aspect ratios between the main current channel and its connecting bridge, which may assume a rectangular shape (for Cartesian channel), and a cylindrical or funnel shape (for cylindrical channel). These scaling laws have been confirmed by spot checks with numerical code results. They are generalizations of the classical theory of Holm and Timsit on the contact resistance of the “$a$-spot,” defined as a small circular area of zero thickness through which current can flow. Potential applications and extensions of the theory are indicated.