Temperature Versus Magnetic Pressure at the Surface of a Semi-Infinite Plate

Transient high magnetic pressures are characteristic of pulsed power applications such as electromagnetic forming, electromagnetic acceleration, and other applications involving high electric currents. Typically, the current pulses are switched on rapidly (microsecond time scale) leading to very high current densities at the surface of the current-carrying conductors due to current-field interactions (skin effect). Electromagnetic diffusion is too slow to enable a spatially homogeneous current distribution inside metal conductors on this time scale. The very high current densities locally generate high ohmic power leading to Joule heating losses and increasing the conductor's surface temperature rapidly. Moreover, high local thermal stresses are induced in the region close to the surface. The combination of magnetically and thermally induced stresses and Joule heating can lead to severe damage of the conductor, including phase transitions and deformations. This paper presents a formula that allows estimating the surface temperature of a semi-infinite conductor being exposed to a transient magnetic pressure at its surface. While this is a textbook problem, if adiabatic conditions are assumed, the approach taken here considers thermal diffusion inside the conducting material. The presented formula is valid if the conductor's physical constants do not depend on temperature and magnetic flux density.