Lorentz Force and Joule Heat Induced in an Electrically Conducting Plate Moving With Time-Dependent Velocity Under the Influence of a Homogeneous Magnetic Field

This paper investigates the interaction of an initially uniform magnetic field with an electrically conducting slab that moves perpendicularly to the magnetic field with arbitrary time-dependent velocity. It is demonstrated that the problem of determining the time-dependent Lorentz force and the time-dependent Joule heat in the slab is mathematically equivalent to solving a 1-D heat diffusion problem with time-dependent boundary conditions and to submitting the solution to a nonstandard postprocessing procedure. For the particular case of an impulsively driven slab we exploit the mathematical analogy between magnetic diffusion and heat diffusion by translating a textbook solution of the corresponding heat-transfer problem into exact and previously unknown relations for Lorentz force and Joule heat. Moreover, we use a 1-D finite-difference code to investigate more general time dependencies of the velocity including smooth accelerations and random velocity changes. Our numerical determination of reaction times (T₉₈) of the Lorentz force in the case of smooth accelerations provides a useful design tool for the development of Lorentz force flowmeters with short reaction times.