Inhomogeneous Electromagnetic Wave Equation - SI Units

SI Units

Maxwell's equations in a vacuum with charge and current sources can be written in terms of the vector and scalar potentials as

\nabla^2 \varphi + {{\partial } \over \partial t} \left ( \nabla \cdot \mathbf{A} \right ) = - {\rho \over \varepsilon_0}
\nabla^2 \mathbf{A} - {1 \over c^2} {\partial^2 \mathbf{A} \over \partial t^2} - \nabla \left ( {1 \over c^2} {{\partial \varphi } \over {\partial t }} + \nabla \cdot \mathbf{A} \right ) = - \mu_0 \mathbf{J}

where

\mathbf{E} = - \nabla \varphi - {\partial \mathbf{A} \over \partial t}

and

\mathbf{B} = \nabla \times \mathbf{A} .

If the Lorenz gauge condition is assumed

{1 \over c^2} {{\partial \varphi } \over {\partial t }} + \nabla \cdot \mathbf{A} = 0

then the nonhomogeneous wave equations become

\nabla^2 \varphi - {1 \over c^2} {\partial^2 \varphi \over \partial t^2} = - {\rho \over \varepsilon_0}
\nabla^2 \mathbf{A} - {1 \over c^2} {\partial^2 \mathbf{A} \over \partial t^2} = - \mu_0 \mathbf{J} .

Read more about this topic:  Inhomogeneous Electromagnetic Wave Equation

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