#### Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations
emf.g.Lg

A more aesthetic choice is the Lorenz gauge:

\[ {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t} = 0 \tag{LorenzG}\label{LorenzG} \] which is chosen to put the second term in the left-hand side of LapA to zero. What remains is then

\[ {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} \tag{ALorenzG}\label{ALorenzG} \] while the equation for \(\phi\) becomes

\[ {\boldsymbol \nabla}^2 \phi - \mu_0 \varepsilon_0 \frac{\partial^2 \phi}{\partial t^2} = -\frac{\rho}{\varepsilon_0}. \tag{phiLorenzG}\label{phiLorenzG} \] These can be written compactly upon introducing a new operator: the

d'Alembertian operator

\[ \square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2} \tag{dAl}\label{dAl} \]

so we get the

Inhomogeneous Maxwell equations (Lorenz gauge)

\[ \square^2 \phi = -\frac{\rho}{\varepsilon_0}, \hspace{10mm} \square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J} \tag{MaxLor}\label{MaxLor} \]

This gauge is especially nice in the context of special relativity. The whole of electrodynamics has thus reduced to solving the inhomogeneous wave equations MaxLor in terms of specified sources.

Without choosing the Lorenz gauge, we can still write the inhomogeneous Maxwell equations in a simpler form. Defining \[ L \equiv {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial \phi}{\partial t}, \] we have by direct inspection \[ \square^2 \phi + \frac{\partial L}{\partial t} = -\frac{\rho}{\varepsilon_0}, \hspace{10mm} \square^2 {\boldsymbol A} - {\boldsymbol \nabla} L = -\mu_0 {\boldsymbol J}. \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42