#### Monochromatic Plane Waves
emd.emw.mpw

A monochromatic wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have \[ {\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm} {\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)} \] Maxwell's equations impose constraints. Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\), \[ (E_0)_z = 0 = (B_0)_z \label{Gr(9.44)} \] so electromagnetic waves are transverse.

From Faraday: \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partial t\), we then get \[ -k(E_0)_y = \omega (B_0)_x, \hspace{1cm} k (E_0)_x = \omega (B_0)_y \Longrightarrow {\bf B}_0 = \frac{k}{\omega} ~\hat{\bf z} \times {\bf E}_0 \label{Gr(9.46)} \] so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and

- Gr (9.47)

\[ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0. \tag{EBmpw}\label{EBmpw} \]

Generalizing to propagation in the direction of an arbitrary wavevector \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the

E and B fields for a monochromatic EM plane wave

- Gr (9.49)

\[ {\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n}, \hspace{10mm} {\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n} = \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t) \tag{mpw}\label{mpw} \] with the transversality condition \[ \hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0 \]

or if you prefer explicit real parts (adding a possible phase shift \(\delta\)): \[ {\boldsymbol E} ({\boldsymbol r},t ) = E_0 \cos({\boldsymbol k} \cdot {\boldsymbol r} - \omega t + \delta) \hat{\boldsymbol n}, \hspace{10mm} {\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} \cos ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t + \delta) ~\hat{\boldsymbol k} \times \hat{\boldsymbol n} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42