#### The Wave Equation
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Take Maxwell's equations in vacuum:

\begin{align} (i) &{\boldsymbol \nabla} \cdot {\bf E} = 0, &(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\ (ii) &{\boldsymbol \nabla} \cdot {\bf B} = 0, &(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = 0, \end{align}

These take the form of coupled first-order partial differential equations for \({\bf E}\) and \({\bf B}\). They can be decoupled: simply take the curl of \((iii)\) and \((iv)\):

\begin{align} {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf E}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf E}) - {\boldsymbol \nabla}^2 {\bf E} = {\boldsymbol \nabla} \times \left( -\frac{\partial {\bf B}}{\partial t} \right) = -\frac{\partial}{\partial t} ({\boldsymbol \nabla} \times {\bf B}) = -\mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2}, \\ {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf B}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf B}) - {\boldsymbol \nabla}^2 {\bf B} = {\boldsymbol \nabla} \times \left(\mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} \right) = \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = -\mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}. \end{align}

Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), we get the

Wave equations for electric and magnetic fields in vacuum

- Gr (9.41)

\[ {\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2}, \hspace{1cm} {\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}. \tag{WaveEq}\label{WaveEq} \]

The equations for the electric and magnetic fields are now decoupled, at the price of becoming second-order equations.

In vacuum, the cartesian components of the fields thus obey the three-dimensional wave equation \[ {\boldsymbol \nabla}^2 f = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}. \] Maxwell's equations therefore support solutions in terms of waves travelling at a speed \[ c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 299 792 458 ~m/s. \] That is, a form \[ {\bf E} ({\bf r},t) = {\bf E}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \hspace{1cm} {\bf B} ({\bf r},t) = {\bf B}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \] solves WaveEq for \(\omega = c |{\bf k}|\). Here and under, we use complex exponentials for convenience, remembering that the actual electric and magnetic fields are given by the real part.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42