Contact resistance of rectangular contact

The conductive body is considered in the form of a parallelepiped, at the ends of which small contacts of the same rectangular shape are connected. The length and the width of these contacts is equal to the values $\varepsilon $ and $\mu$, considered below as small parameters. The case of a uniform current density at the contacts is considered. A physical situation close to it occurs, for example, in the presence of a thin, poorly conductive film on the contact surface. The potential of the electric current of the sample is modeled with the help of a solution for the Neumann problem to the Laplace equation in a parallelepiped. The electrical resistance is calculated as the sum of a double series, singularly depending on two small parameters $\varepsilon$ and $\mu$. We consider the case when $\mu=k\varepsilon$, where $k$ is some constant. The principal term of the asymptotic expansion of the sum of the given series for $\varepsilon\to0$ corresponds to the contact resistance of a rectangular contact with the sides $2\varepsilon$ and $2\mu$. The purpose of this paper is to obtain an explicit expression for this contact resistance.