Near and far field

The near and far field distinguish source-dependent electromagnetic field regions. Close to a localized source, reactive fields and induction terms contribute; sufficiently far away, the leading transverse fields decrease as the inverse distance and carry outgoing power. The radiation zone describes this far-field limit.

For a harmonic magnetic dipole observed with kr of order unity, its quasistatic, induction and radiation fields scale as $r^{-3}$, $k{r}^{-2}$ and $k^2{r}^{-1}$ and need not be cleanly separated experimentally. The radiated-power law remains an asymptotic flux statement, but $P/(4\pi r^2)$ alone does not specify the local detected field. Detectability further depends on directional pattern, plasma propagation, antenna coupling, bandwidth, integration and noise. A tiny nonzero vacuum radiation term does not by itself establish either a practical detection or an absolute impossibility.

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 $E_0=\sqrt{2⟨S⟩/({ϵ}_{0}c)}$.

Multipole radiation expands radiation from localized time-dependent sources into electric and magnetic multipoles. Electric dipole radiation is the leading electric term, while magnetic dipole radiation and higher electric multipoles contribute when the leading term vanishes or higher precision is needed.

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

With all electric dipole moments and time derivatives evaluated at $t_0=t-r/c$, the retarded potentials of a point electric dipole give, for $r>0$,For a short verification, apply $\nabla t_0=-\mathbf{n}/c$ and ${\partial }_i{n}_j=({\delta }_{ij}-n_i{n}_j)/r$ when computing $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ and $\mathbf{B}=\nabla \times \mathbf{A}$. This produces the displayed $r^{-3}$, $r^{-2}$ and $r^{-1}$ terms. The radiation electric field is transverse and obeys ${\mathbf{B}}_{\mathrm{rad}}=\mathbf{n}\times {\mathbf{E}}_{\mathrm{rad}}/c$. The vector triple product then gives the leading Poynting vector $\mathbf{n}[|{\mathbf{p}}^{\prime \prime }{|}^2-(\mathbf{n}\cdot {\mathbf{p}}^{\prime \prime }{)}^2]/(16{\pi }^2{ϵ}_0{c}^3{r}^2)$, displaying the angular distribution of electric dipole radiation.

A permanent electric dipole moment of magnitude $p_0$ rotating perpendicular to a fixed axis emits $P=K{\omega }^4$ in the electric-dipole approximation for radiation, with $K={\mu }_0{p}_0^2/(6\pi c)$. For fixed mechanical moment of inertia $\mathcal{I}$, weak radiative loss gives $\mathcal{I}\omega \dot{\omega }=-K{\omega }^4$ and hence the displayed spin-down law. It assumes $|\dot{\omega }|\ll {\omega }^2$: the full dipole acceleration has squared magnitude $p_0^2({\omega }^4+{\dot{\omega }}^2)$, whose second term is beyond this leading adiabatic approximation.

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.

The angular distribution of electric dipole radiation measures squared projections of the dipole axis, so a few intensity measurements need not determine that axis or the total radiated power uniquely. For a real unit dipole axis $\mathbf{d}$, the radiation-zone formula is $I(\mathbf{n})=3P[1-(\mathbf{d}\cdot \mathbf{n}{)}^2]/(8\pi R^2)$. Equal intensities along $\hat{\mathbf{z}}$, $(\hat{\mathbf{x}}+\hat{\mathbf{z}})/\sqrt{2}$ and $(\hat{\mathbf{y}}+\hat{\mathbf{z}})/\sqrt{2}$ permit four axes, up to reversal:Three generally different radiated powers are compatible with the same intensity data. Additional directional measurements or polarization information can remove the ambiguity.

For an orbit of radius $b$ in the $xy$ plane, $\mathbf{a}=-b{\omega }^2(\cos \omega t,\sin \omega t,0)$. In spherical coordinates, at angle $\theta$ from the orbit normal, $⟨(\mathbf{n}\cdot \mathbf{a}{)}^2⟩=b^2{\omega }^4{\sin }^2\theta /2$. The acceleration covariance in nonrelativistic radiation therefore gives the displayed pattern. Its axial power per solid angle is twice its equatorial value; no observation direction has zero time-averaged power. Integrating gives $⟨P⟩=q^2{b}^2{\omega }^4/(6\pi {ϵ}_0{c}^3)$. The condition $b\omega \ll c$ is essential: relativistic synchrotron radiation has a different beamed pattern.

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

This is the magnetic-dipole term of the multipole radiation expansion.

Let a conducting loop of area A rotate about z with normal $(\cos \omega t,\sin \omega t,0)$ in a fixed field $B\hat{\mathbf{x}}$. Neglect inductance so $I=(BA\omega /R)\sin \omega t$. Put $m_0=B{A}^2\omega /R$. Its vector magnetic moment isThe static component does not radiate. The remaining component rotates at twice the mechanical frequency and has $|\ddot{\mathbf{m}}{|}^2=4m_0^2{\omega }^4$. Thus $\overline{P}=2{\mu }_0{B}^2{A}^4{\omega }^6/(3\pi c^3{R}^2)$. Treating the same sinusoidal current as a fixed-axis moment underestimates this rotating geometry by a factor of eight. The energy source is mechanical rotation, with additional ohmic dissipation and magnetic braking.

For constant self-inductance L, the steady induced current satisfies $L\dot{I}+RI=BA\omega \sin \omega t$, giving amplitude $I_0=BA\omega /\sqrt{R^2+{\omega }^2{L}^2}$ and phase lag. Multiplication by the rotating loop normal again gives a static moment plus a twice-frequency rotating moment of magnitude $A{I}_0/2$, independently of lag. Therefore replace the resistance-only denominator $R^2$ in the radiation power by $R^2+{\omega }^2{L}^2$. A wire resistance alone is not sufficient for an exact prediction if inductance is appreciable.

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