= Electromagnetic radiation
{wiki}
<Electromagnetic radiation> is a propagating disturbance of the <electric field> and <magnetic field> that carries <energy> and <momentum> away from its source. It can be a sustained <electromagnetic wave> or a finite <wave packet>: periodic oscillation is not required. In vacuum the disturbance travels at the <speed of light>. A receiver responds to fields arriving now from an earlier source event, as expressed by <retarded time>. This is how electrical activity at one place can deliver <energy> and information elsewhere without a connecting wire.
Stored <electromagnetic energy> alone does not establish outgoing <electromagnetic radiation>. A stationary <electric charge> in <electrostatics>, or a steady localized <electric current> in <magnetostatics>, can store <electromagnetic energy> with zero <Poynting vector>. Static combinations can have circulating <energy flux> without net outward <power>. Near a changing source, reactive <energy> can return later. The <near and far field> distinction separates these contributions from outgoing <radiation zone> fields.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others/Chapter 27: Field Energy and Field Momentum>[§27-5, Fig. 27-6: static fields with circulating energy flux]{ref}
A single traveling vacuum <plane electromagnetic wave> obeying the <Maxwell equations> has $\mathbf B=\widehat{\mathbf k}\times\mathbf E/c$. Its <Poynting vector> gives the <energy density and flux of a plane electromagnetic wave>:
$$
\mathbf S=\frac{\mathbf E\times\mathbf B}{\mu_0}
=\epsilon_0c|\mathbf E|^2\widehat{\mathbf k},\qquad
I=\langle\mathbf S\rangle\cdot\widehat{\mathbf k}
=\frac{c\epsilon_0E_0^2}{2}.
$$
Here $I$ is <wave intensity>; $\epsilon_0$ and $\mu_0$ are <vacuum permittivity> and <vacuum permeability>. For a <monochromatic wave> with <linear polarization> and peak amplitude $E_0$, squared <cosine> averages to $1/2$. Pulses require the instantaneous <Poynting vector>; arbitrary incident-and-reflected superpositions need not obey this relation.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others/Chapter 27: Field Energy and Field Momentum>[§27-5, Eqs. (27.16)–(27.18): traveling-wave energy flux]{ref}
The <electromagnetic momentum density> is $\mathbf g=\mathbf S/c^2$. Completely absorbing a unidirectional vacuum pulse of <energy> $U$ transfers <momentum> $U/c$; see <radiation pressure on a perfectly absorbing surface>. A calculated $1.0\,\mathrm{mJ}$ pulse transfers $3.34\times10^{-12}\,\mathrm{kg\,m/s}$. Symmetric emission can lose <energy> without net recoil because opposite <momenta> cancel.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others/Chapter 27: Field Energy and Field Momentum>[§27-6, Eq. (27.21) and Fig. 27-7: field momentum and absorbed energy]{ref}
The direction of emission matters as much as its total <power>. For an oscillating <electric dipole moment> along $z$, the <angular distribution of electric dipole radiation> is strongest across the axis and vanishes along it. Its origin is geometric: the <electric dipole radiation> field is the transverse projection of the second time derivative of the <electric dipole moment>, so its magnitude contains $\sin\theta$. Squaring this <electric field> to obtain <wave intensity> gives $\sin^2\theta$. The <electric-dipole approximation for radiation> requires a source much smaller than its <wavelength> and observation in the <radiation zone>. <The Feynman Lectures on Physics, Volume I by Richard Feynman and others/Chapter 28: Electromagnetic Radiation>[Feynman, Leighton and Sands, Electromagnetic Radiation, §§28-2–28-3: transverse acceleration and dipole geometry]{ref}
\Image[/electric-dipole-radiation-angular-pattern.png]
{title=Normalized angular pattern of <electric dipole radiation>}
{description=Calculated <radiation zone> model, not measured data. The polar curve's radial coordinate is normalized <power> per <solid angle>, $\sin^2\theta$, rather than physical distance or a field line. The angle $\theta$ is measured from the dipole axis. Rotating this meridional pattern about that axis gives the three-dimensional angular distribution.}
For example, the <power> per <solid angle> at $\theta=30^\circ$ is one quarter of its equatorial value, while the <electric field> amplitude is one half. To normalize the angular distribution to total <power>, use <spherical coordinates> with $d\Omega=\sin\theta\,d\theta\,d\phi$:
$$
\int\sin^2\theta\,d\Omega
=2\pi\int_0^\pi\sin^3\theta\,d\theta=\frac{8\pi}{3},\qquad
\frac{1}{P}\frac{dP}{d\Omega}=\frac{3}{8\pi}\sin^2\theta.
$$
The <integral> follows by substituting $x=\cos\theta$, giving $\int_{-1}^1(1-x^2)\,dx=4/3$. This is a probability-like normalization over direction, with units of inverse <steradian>; it does not make the dipole an isotropic radiator. The same angular <integral> underlies the <Larmor formula> for nonrelativistic accelerated <electric charge>.
Detection requires the incident <electromagnetic field> to do something observable. An <antenna> develops an electrical response; a <thermopile> converts absorbed <energy> into a thermal and then electrical signal; a <photodetector> can respond through <photon> absorption. <Hertz spark transmission and detection experiment> makes this conversion visible as a tiny spark. Detector orientation, sensitivity, and surrounding reflections all matter: failure to see a signal does not prove that no <electromagnetic radiation> reached the receiver.
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