# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

#### Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations
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A more aesthetic choice is the {\bf Lorenz gauge}: \[ {\boldsymbol \nabla} \cdot {\boldsymbol A} + \mu_0 \varepsilon_0 \frac{\partial V}{\partial t} = 0 \label{eq:LorenzGauge} \] which is chosen to put the second term in the left-hand side of (\ref{eq:LaplacianA}) to zero. What remains is then \[ {\boldsymbol \nabla}^2 {\boldsymbol A} - \mu_0 \varepsilon_0 \frac{\partial^2 {\boldsymbol A}}{\partial t^2} = -\mu_0 {\boldsymbol J} \] while the equation for \(V\) becomes \[ {\boldsymbol \nabla}^2 V - \mu_0 \varepsilon_0 \frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0}. \] These can be written compactly upon introducing a new operator: the

{\bf d'Alembertian operator} \[ \square^2 \equiv {\boldsymbol \nabla}^2 - \mu_0 \varepsilon_0 \frac{\partial^2}{\partial t^2} \label{eq:dAlembertian} \]

so we get the

{\bf Inhomogeneous Maxwell equations (Lorenz gauge)} \[ \square^2 V = -\frac{\rho}{\varepsilon_0}, \hspace{10mm} \square^2 {\boldsymbol A} = -\mu_0 {\boldsymbol J} \label{eq:InhomogeneousMaxwellLorenzGauge} \]

This gauge is especially nice in the context of special relativity. The whole of electrodynamics has thus reduced to solving the inhomogeneous wave equations (\ref{eq:InhomogeneousMaxwellLorenzGauge}) in terms of specified sources.

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

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Author: Jean-Sébastien Caux

Created: 2022-02-09 Wed 22:40