Larmor formula

The Larmor formula , after Joseph Larmor , is a formula from classical electrodynamics , from which the radiated power of an accelerated, electrically charged particle can be calculated. It follows from the Liénard-Wiechert potentials , which determine the electromagnetic field of an accelerated charge, and Poynting's theorem , which transfers the law of conservation of energy to electrodynamics. The fact that power is radiated follows directly from the law of conservation of energy: If a particle loses energy in an electromagnetic field, it must be emitted in the form of electromagnetic radiation.

Classic borderline case

In a non-relativistic approximation, i.e. when the speed of the particle relative to the observer is small compared to the speed of light, the Larmor formula for the differential power per solid angle element is :

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \Omega }}={\frac {e^{2}}{16\pi ^{2}\varepsilon _{0}c}}({\dot {\vec {\beta }}})^{2}\sin ^{2}\theta }$

Where:

• ${\displaystyle P}$ the performance
• ${\displaystyle \Omega }$ the solid angle
• ${\displaystyle \varepsilon _{0}}$the electric field constant
• ${\displaystyle e}$the elementary charge
• ${\displaystyle c}$the speed of light
• ${\displaystyle {\vec {\beta }}}$the speed in units of the speed of light ( ) ${\displaystyle {\vec {\beta }}={\vec {v}}/c}$
• ${\displaystyle \theta }$ the angle between the acceleration vector and the observation point

The total radiated power results as an integration over the solid angle to:

${\displaystyle P={\frac {e^{2}}{6\pi \varepsilon _{0}c}}({\dot {\vec {\beta }}})^{2}}$

Relativistic generalization

A relativistic calculation results

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \Omega }}={\frac {e^{2}}{16\pi ^{2}\varepsilon _{0}c}}{\frac {\left|{\vec {n}}\times [({\vec {n}}-{\vec {\beta }})\times {\dot {\vec {\beta }}}]\right|^{2}}{(1-{\vec {n}}\cdot {\vec {\beta }})^{5}}}}$

with the unit vector between the observation point and the location of the charge. In the relativistic case, the power is to be understood as a change in the energy per proper time interval . The integration over the solid angle results ${\displaystyle {\vec {n}}}$

${\displaystyle P={\frac {e^{2}}{6\pi \varepsilon _{0}c}}\gamma ^{6}\left(({\dot {\vec {\beta }}})^{2}-({\vec {\beta }}\times {\dot {\vec {\beta }}})^{2}\right)}$

with the Lorentz factor . ${\displaystyle \gamma }$

literature

• John David Jackson: Classical Electrodynamics . 3. Edition. John Wiley & Sons, Hoboken 1999, ISBN 0-471-30932-X (English).