Structure of a plane wave

A plane electromagnetic wave propagating along the $x$ axis has a very simple, regular structure. The electric field oscillates along a fixed direction (say $y$), the magnetic field along an orthogonal direction ($z$), and both are perpendicular to the direction of propagation: the wave is transverse.

Plane harmonic electromagnetic wave

$\vec{E} = \left(0,\; E_0\cos(kx-\omega t),\; 0\right)$ $\vec{B} = \left(0,\; 0,\; B_0\cos(kx-\omega t)\right)$ with $k = 2\pi/\lambda$, $\omega = 2\pi/T$, and the relations $\lambda/T = c \qquad\text{and}\qquad |B_0| = |E_0|/c$

The two fields oscillate in phase (same $\cos(kx-\omega t)$): they vanish together and reach their maximum together. The vectors $\vec{E}$, $\vec{B}$ and the direction of propagation form a right-handed triad, orderable with the right-hand rule: $\vec{E}\times\vec{B}$ points in the direction of propagation.

Plane EM wave: $\vec{E}$ (gold) oscillates in the $xy$ plane, $\vec{B}$ (blue) in the $xz$ plane, in phase; the wave advances along $x$ at speed $c$.

The fact that $|B_0| = |E_0|/c$ explains why the magnetic field of a light wave is numerically tiny compared with the electric field: dividing by $c \approx 3\cdot 10^8$ m/s reduces it by almost nine orders of magnitude. This does not mean that $\vec{B}$ is “less important”: the two fields carry, on average, the same energy.

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Topics: Electromagnetic waves Concepts: Electromagnetic wave · Harmonic wave · Electric field · Magnetic field

Related exercises: Problem — Amplitudes of the solar wave · Problem — Ratio E0 to B0 · Problem — Microwave oven