  We estimate the temperature distribution in the rails of an electromagnetic
rail gun (EMG) due to the confinement of the current in a narrow surface layer
resulting from the skin effect. In order to obtain analytic results, we assume
a simple geometry for the rails, an electromagnetic skin effect boundary edge
that propagates with the accelerating armature, and a current carrying channel
controlled by magnetic field diffusion into the rails. We compute the
temperature distribution in the rails at the time that the armature leaves the
rails. For the range of exit velocities, from 1500 m/s to 5000 m/s, we find the
highest temperatures are near the gun breech. After a single gun firing, the
temperature reaches the melting temperature of the metal rails in a layer of
finite thickness near the surface of the rails, for rails made of copper or
tantalum. We plot the thickness of the melt layer as a function of position
along the rails. In all cases, the thickness of the melt layer increases with
gun velocity, making damage to the gun rails more likely at higher velocity. We
also calculate the efficiency of the EMG as a function of gun velocity and find
that the efficiency increases with increasing velocity, but only if the length
of the gun is sufficiently long. The thickness of the melted layer also
decreases with increasing rail length. Therefore, there is a tradeoff: for
rails of sufficient length, the gun efficiency increases with increasing
velocity but the melted layer thickness in the rails also increases.
