Electromagnetic wave

An electromagnetic wave is a propagating coupled oscillation of electric and magnetic fields governed by Maxwell's equations.

The refractive index is the ratio of the vacuum speed of light to a medium's phase speed. In a homogeneous isotropic medium, $n=c\sqrt{\epsilon \mu }$ in compatible units.

In vacuum, each field obeys ${\nabla }^{2}E-c^{-2}{E}_{tt}=0$ and ${\nabla }^{2}B-c^{-2}{B}_{tt}=0$.

For a vacuum plane wave, $k\cdot E_0=0$, $\omega =c|k|$, and $B_0=k\times E_0/\omega$.

The polarization of an electromagnetic wave describes the path traced by its transverse electric-field vector at a fixed point. A monochromatic wave may be linearly, circularly, or elliptically polarized; in vacuum it has two independent transverse polarization states.

The Stokes parameters $I,Q,U,V$ describe intensity, two linear-polarization components and circular polarization. $Q$ and $U$ depend on the chosen transverse basis. On the celestial sphere, $Q\pm iU$ transform as spin-two fields rather than as ordinary scalars.

A wave is linearly polarized when its oscillating electric field always lies along one fixed line. For a transverse plane wave, that line is perpendicular to the wavevector. Unlike circular polarization, the field oscillates along the line rather than rotating through directions.

An electromagnetic wave has circular polarization when its transverse electric field has constant magnitude and rotates once during each period. In complex-amplitude notation, two orthogonal components have equal magnitude and a relative phase of one quarter-cycle.

At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

At normal incidence, the reflected electric amplitude is the negative of the incident amplitude at the surface, while the reflected magnetic amplitude has the same sign. Their superposition forms a standing wave with an electric node and a magnetic antinode at the conductor.

Without free surface charge or current, the tangential components of $E$ and $H$ and the normal components of $D$ and $B$ are continuous across an interface.

Equality of the boundary fields for every tangential position and time forces equal frequency and equal tangential wavevector for incident, reflected, and transmitted plane waves.

For refractive indices $n_{-}$ and $n_{+}$,

Total internal reflection occurs when a wave travels from higher to lower refractive index above the critical incidence angle, leaving no propagating transmitted plane wave normal to the interface.

For refractive indices $n_{-}>n_{+}$, the critical angle satisfies $\sin {\theta }_c=n_{+}/n_{-}$. Above it, the transmitted normal wavenumber is imaginary and the transmitted field is an evanescent wave.

When the permeabilities agree, the signed reflected-to-incident electric-field ratio for polarization normal to the plane of incidence isFor incidence from lower to higher refractive index it never vanishes.

For equal permeabilities on the two sides of an interface, tangential-field continuity and Snell law for electromagnetic waves givefor consistently signed electric amplitudes normal to the plane of incidence.

For $D_i={\epsilon }_{ij}{E}_j$, $B=\mu H$, and a plane wave with electric polarization $e$, the source-free Maxwell equations implyIf $e$ is an eigenvector of $\epsilon$, then $e\perp k$, the dispersion relation is $|k{|}^2={\omega }^2\mu {\epsilon }_e$, and the Poynting vector is parallel to $k$.

The Poynting vector $S=E\times B/{\mu }_0$ is electromagnetic energy flux.

For electromagnetic energy densityMaxwell's equations implyThe field energy lost from a volume becomes outward electromagnetic flux or work on charges.

In vacuum the electromagnetic energy density isand the field energy in a region is $U_{\mathrm{EM}}=\int udV$.

In vacuum-permeability magnetohydrodynamics, the magnetic energy in a volume is $E_B=\int B^2/(2{\mu }_0)dV$. Its volume density equals the magnetic pressure.

For a vacuum plane wave, $B=\hat{k}\times E/c$, soThe electric and magnetic energy densities are equal, and the time averages satisfy $⟨S⟩=c⟨u⟩\hat{k}$.

For symmetric time-independent constitutive tensors,and the macroscopic Maxwell equations give ${\partial }_{t}u+\nabla \cdot S=-E\cdot J$.

In the radiation zone of a localized source, the leading electromagnetic fields decay as $1/R$, are transverse to the observation direction, and satisfy $B=\hat{R}\times E/c$. Terms decaying faster than $1/R$ do not contribute to the limiting radiated power.

An isotropic source of average power $P$ has intensityat distance $r$. For monochromatic vacuum radiation, its electric-field amplitude is $E_0=\sqrt{2⟨S⟩/({ϵ}_{0}c)}$.

For a localized source whose size is small compared with its radiation wavelength, the radiation fields of its electric dipole moment $p$ areTheir total instantaneous power is

For a source of characteristic size $a$ and angular frequency $\omega$, the electric-dipole approximation requiresso the source is much smaller than its radiation wavelength. Observing the pure $1/R$ radiation field additionally requires the radiation zone, $R\gg c/\omega$ and $R\gg a$.

If $\theta$ is the angle between the observation direction and $\ddot{p}$, thenIntegrating over solid angle uses $\int {\sin }^2\theta d\Omega =8\pi /3$ and gives the total electric dipole radiation power.

A charge $q$ undergoing simple harmonic motion of angular frequency $\omega$ and slowly varying amplitude $A$ radiates the averaged powerIts mechanical energy $E=m{A}^2{\omega }^2/2$ therefore has exponential decay rate $\gamma ={\mu }_0{q}^2{\omega }^2/(6\pi mc)$ and half-life

Electromagnetic duality replaces the electric dipole $p$ by $m/c$ for a magnetic dipole moment $m$. Therefore

If a rigid body's inclined magnetic dipole rotates at angular speed $\Omega$, its radiated power is $C{\Omega }^4$. Conservation of rotational energy $I{\Omega }^2/2$ then gives