New distributional Laplacian identities for arsinh[a(z-b)/s] and ln[a(z-b)+sqrt(s^2+a^2(z-b)^2)] are derived and applied to solve the Coulomb-gauge Poisson equation for a uniformly moving charge.
Potentials and fields of a charge set suddenly from rest into uniform motion
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abstract
The fact that electromagnetic effects propagate at the speed of light suggests how the Lorenz-gauge scalar and vector potentials of a uniformly moving point charge must be modified when the charge was initially at rest and then set suddenly into uniform motion. The modified potentials are shown to satisfy the requisite inhomogeneous wave equations. The gauge function of the transformation of these potentials to the Coulomb gauge is calculated in closed form. It is validated by confirming that the Coulomb-gauge vector potential that is calculated using it yields together with the Coulomb-gauge scalar potential the same electric and magnetic fields as those calculated with the Lorenz-gauge potentials.
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Novel distributional Laplacians and a Coulomb-gauge problem
New distributional Laplacian identities for arsinh[a(z-b)/s] and ln[a(z-b)+sqrt(s^2+a^2(z-b)^2)] are derived and applied to solve the Coulomb-gauge Poisson equation for a uniformly moving charge.