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Mathematics for Physics

80 Radiation

Electromagnetic energy that leaves a source forever that is used to describe fields that detach from accelerating charges and travel outward as waves.

The radiation criterion. Radiation requires the outward energy flux to fall no faster than \(1/r^{2}\) at large distance. This principle is used to tell true radiating fields apart from near fields that keep energy near the charges.

The radiated power is

\[ P_{\mathrm{rad}}(t) = \lim_{r\to\infty}\oint\mathbf{S}\cdot d\mathbf{a} \]

where

  • \(P_{\mathrm{rad}}\) is the radiated power.
  • \(\mathbf{S}\) is the Poynting vector.
  • \(d\mathbf{a}\) is the outward vector area element.
  • \(r\) is the distance from the source.

The Larmor formula. Only accelerating charges radiate. This principle is used to compute the power lost by a nonrelativistic accelerating charge.

The Larmor formula is

\[ P = \dfrac{\mu_{0}q^{2}a^{2}}{6\pi c} \]

where

  • \(P\) is the radiated power.
  • \(q\) is the charge.
  • \(a\) is the magnitude of the acceleration.
  • \(\mu_{0}\) is the permeability of free space.
  • \(c\) is the speed of light.

The radiation-zone \(1/r\) fields. In the radiation zone the surviving fields fall as \(1/r\) and are transverse to the travel direction. The radiation zone is the region much farther than both the source size and the wavelength. This principle is used to write the fields that a distant receiver measures.

The radiation-zone amplitude scales as

\[ E,\,B \propto \dfrac{1}{r} \]

where

  • \(E\) is the electric field magnitude.
  • \(B\) is the magnetic field magnitude.
  • \(r\) is the distance from the source.