Inductance calculation of a coil gun that launches a thin plate edge-on

A coil gun is a type of induction launcher that induces an eddy current in a metal projectile by a time varying magnetic induction produced by a launch coil. The interaction between the magnetic induction of the coil and the eddy currents can launch the plate with the velocity vector in the plane of the plate, i.e. edge-on. A previous published calculation of the eddy current in thin plates includes the conductivity of the plate and the time variation of the applied magnetic field, which complicates the problem. Although these features are necessary for a general treatment of the subject, they complicate the problem and may not be necessary for applications where the skin depth is small compared to the length or width of the plate. For simplification, it has been assumed that the rectangular plate has an infinite conductivity, the plate has zero thickness, the applied magnetic induction is static, and the streamline function for the eddy currents can be expressed as a polynomial with adjustable coefficients. The curl of the stream line function yields the current density distribution. Under these assumptions, the integrals that result from the application of the Biot-Savart law are in principle analytic for any order of the polynomial, but their functional forms are very complicated. Their values, however, can be found by using recursion relations. After the evaluation of these integrals, the coefficients of the polynomial are adjusted so that the boundary conditions for the magnetic induction are satisfied on the plate's surface. This streamline function is then used to find the mutual inductance between the launch coil and plate for the given position. This procedure is repeated for other plate positions. With these results, it is possible to design the pulsed power supply for the coil gun and to predict the plate's velocity.