In the radiation zone of a localized source, the leading electromagnetic fields decay as , are transverse to the observation direction, and satisfy . Terms decaying faster than do not contribute to the limiting radiated power.
An isotropic source of average power has intensityat distance . For monochromatic vacuum radiation, its electric-field amplitude is .
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 areTheir total instantaneous power is
With all electric dipole moments and time derivatives evaluated at , the retarded potentials of a point electric dipole give, for ,For a short verification, apply and when computing and . This produces the displayed , and terms. The radiation electric field is transverse and obeys . The vector triple product then gives the leading Poynting vector , displaying the angular distribution of electric dipole radiation.
A permanent electric dipole moment of magnitude rotating perpendicular to a fixed axis emits in the electric-dipole approximation for radiation, with . For fixed mechanical moment of inertia , weak radiative loss gives and hence the displayed spin-down law. It assumes : the full dipole acceleration has squared magnitude , whose second term is beyond this leading adiabatic approximation.
For a source of characteristic size and angular frequency , the electric-dipole approximation requiresso the source is much smaller than its radiation wavelength. Observing the pure radiation field additionally requires the radiation zone, and .
If is the angle between the observation direction and , thenIntegrating over solid angle uses 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 , the radiation-zone formula is . Equal intensities along , and 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 in the plane, . In spherical coordinates, at angle from the orbit normal, . 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 . The condition is essential: relativistic synchrotron radiation has a different beamed pattern.
A charge undergoing simple harmonic motion of angular frequency and slowly varying amplitude radiates the averaged powerIts mechanical energy therefore has exponential decay rate and half-life
This is the magnetic-dipole term of the multipole radiation expansion.
Let a conducting loop of area A rotate about z with normal in a fixed field . Neglect inductance so . Put . Its vector magnetic moment isThe static component does not radiate. The remaining component rotates at twice the mechanical frequency and has . Thus . 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 , giving amplitude and phase lag. Multiplication by the rotating loop normal again gives a static moment plus a twice-frequency rotating moment of magnitude , independently of lag. Therefore replace the resistance-only denominator in the radiation power by . 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 , its radiated power is . Conservation of rotational energy then gives
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The radiation zone is a region within a star, specifically in stars like the Sun, where energy produced in the core is transported outward primarily through radiation. This zone is found between the inner core, where nuclear fusion occurs, and the outer convective zone. In the radiation zone, the high temperature and pressure conditions cause photons (light particles) to be absorbed and re-emitted by particles in the plasma, leading to a slow transfer of energy outwards.