Electromagnetic wave

Electromagnetic fields (that is, vector fields) propagate as waves. From Maxwell’s equations, we can do a few manipulations (curl, divergence, and substitutions), we get the following:

$${\nabla }^2\vec{E}=\mu ϵ\frac{{\partial }^2\vec{E}}{\partial t^2}$$ $${\nabla }^2\vec{H}=\mu ϵ\frac{{\partial }^2\vec{H}}{\partial t^2}$$

From the 3D wave equation, we can conclude that EM fields can propagate as waves with speed:

$$c=\frac{1}{\sqrt{\mu ϵ}}$$

where $c$ is the speed of light (i.e., light is an electromagnetic wave).

Harmonic plane wave

We can express an EM wave as a harmonic plane wave. The main important parameter is $\vec{k}$, which describes the direction of propagation.

$$\vec{E}(\vec{r},t)={\vec{E}}_o\exp \left[i\left(\vec{k}\cdot \vec{r}\pm \omega t+{\phi }_o\right)\right]$$

While we usually express using the electric field, a propagating EM wave has an accompanying magnetic field portion. We make two important substitutions for operations, the gradient and partial derivative with respect to time.

$$\nabla ⟷i\vec{k}$$ $$\frac{\partial }{\partial t}⟷-i\omega$$

Additionally, $\vec{E},\vec{H},\vec{k}$ are all mutually perpendicular. They form a right-hand triplet, and are associated by:

$$\vec{k}\times \vec{E}=\mu \omega \vec{H}$$