Larmor formula

A Yagi–Uda antenna. Radio waves can be radiated from an antenna by accelerating electrons in the antenna. This is a coherent process, so the total power radiated is proportional to the square of the number of electrons accelerating.

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897,^[1] in the context of the wave theory of light.

When any charged particle (such as an electron, a proton, or an ion) accelerates, energy is radiated in the form of electromagnetic waves. For a particle whose velocity is small relative to the speed of light (i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula:

P={\frac {2}{3}}{\frac {q^{2}}{4\pi \varepsilon _{0}c}}\left({\frac {\dot {v}}{c}}\right)^{2}={2 \over 3}{\frac {q^{2}a^{2}}{4\pi \varepsilon _{0}c^{3}}}={\frac {q^{2}a^{2}}{6\pi \varepsilon _{0}c^{3}}}=\mu _{0}{\frac {q^{2}a^{2}}{6\pi c}}{\text{ (SI units)}}

P={\frac {2}{3}}{\frac {q^{2}a^{2}}{c^{3}}}{\text{ (cgs units)}}

where

{\dot {v}}

or

a

is the proper acceleration,

q

is the charge, and

c

is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials.

In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as:

P={\frac {2}{3}}{\frac {m_{e}r_{e}a^{2}}{c}}

One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced.

^ Larmor, J (1897). "LXIII.On the theory of the magnetic influence on spectra; and on the radiation from moving ions". Philosophical Magazine. 5. 44 (271): 503–512. doi:10.1080/14786449708621095. Formula is mentioned in the text on the last page.

[1] Larmor, J (1897). "LXIII.On the theory of the magnetic influence on spectra; and on the radiation from moving ions". Philosophical Magazine. 5. 44 (271): 503–512. doi:10.1080/14786449708621095. Formula is mentioned in the text on the last page.

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