# Electromagnetic Waves

Electromagnetic waves consists of oscillating electric and magnetic fields that propagate through space at the speed of light (3x108 m/s). Electromagnetic waves do not require a medium to travel through, and can travel through a vacuum. These waves are produced by the acceleration charges. EM waves include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

## EM waves in free space

In free space (i.e., in the absence of any charges or currents), Maxwell's equations take the form \begin{align} \nabla\cdot\vec{E}=0, \\ \nabla\cdot\vec{B}=0, \\ \nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}, \\ \nabla\times\vec{B}=\frac{1}{c^2}\frac{\partial\vec{E}}{\partial t}, \end{align} where \begin{align} c=\sqrt{\frac{1}{\mu_0\epsilon_0}}. \end{align}

Maxwell's equations gives wave equations for the electric and magnetic fields \begin{align} \nabla^2\vec{E}-\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}=0, \\ \nabla^2\vec{B}-\frac{1}{c^2}\frac{\partial^2\vec{B}}{\partial t^2}=0. \end{align}

The solution of these wave equations are plane electromagnetic waves. The electric and magnetic fields in a plane electromagnetic wave at position $\vec{r}$ and time $t$ are given by \begin{align} \vec{E}(\vec{r},t)&=\vec{E}_0\cos\left(\vec{k}\cdot\vec{r}-\omega t+\phi_0\right),\\ \vec{B}(\vec{r},t)&=\vec{B}_0\cos\left(\vec{k}\cdot\vec{r}-\omega t+\phi_0\right), \end{align} where $\vec{E}_0$ is a constant vector, $\phi_0$ is a constant phase, $\omega$ and $\vec{k}$ are constants related to the frequency $\nu$ and wavelength $\lambda$ by \begin{align} \omega&=2\pi\nu,\\ \lambda&=\frac{2\pi}{|\vec{k}|}=\frac{2\pi}{k}. \end{align} The vectors $\vec{E}_0$ and $\vec{B}_0$ are the amplitudes of the electric and magnetic fields. The wave is travelling in the direction $\hat{k}$ with a speed $\omega/k$.

Consider a plane EM wave travelling in $z$-direction. If E-field is along $x$ direction then expressions for the fields are \begin{align} \vec{E}(z,t)&=\vec{E}_0\hat\imath\cos\left(kz-\omega t\right),\\ \vec{B}(z,t)&=\vec{B}_0\hat\jmath\cos\left(kz-\omega t\right). \end{align}

### Relation between E and B fields

In an EM wave the electric field $\vec{E}$, the magnetic field $\vec{B}$ and the direction of propagation $\vec{k}$ are perpendicular to each other. In mathematical terms, \begin{align} \vec{k}\cdot\vec{E}_0=0, \\ \vec{k}\cdot\vec{B}_0=0, \end{align} and \begin{align} \vec{k}\times\vec{E}_0=\omega\vec{B}_0, \\ \vec{k}\times\vec{B}_0=-\omega\vec{E}_0. \end{align} The speed of the wave is related to the fields by \begin{align} c=\sqrt{\frac{1}{\mu_0\epsilon_0}}=\frac{E}{B}. \end{align}

### Energy of EM wave

Consider a plane electromagnetic wave. The energy density in the electric field is given by \begin{align} U_E & =\frac{1}{2}\epsilon_0E^2 \\ &=\frac{1}{2}\epsilon_0E_0^2\cos^2\left(\vec{k}\cdot\vec{r}-\omega t\right) \end{align} The average energy density in electric field (take the average over time at any point in space) is \begin{align} \bar{U}_E=\frac{1}{4}\epsilon_0E_0^2. \end{align} The average energy density in magnetic field is \begin{align} \bar{U}_E=\frac{1}{4}\frac{B_0^2}{\mu_0}. \end{align} The average energy density in EM wave is \begin{align} \bar{U}=\bar{U}_E+\bar{U}_B=\frac{1}{2}\epsilon_0 E_0^2. \end{align}

The energy flux is the energy crossing per unit area (perpendicular to the direction of propagation) per unit time. The energy flux of EM waves is given by \begin{align} S=Uc=\frac{1}{2}\epsilon_0c E_0^2. \end{align}

or