# What is the relation between the frequency of the light produced and the acceleration of the charged particle

We know that accelerating charges produce EM radiation. Can we derive a relation between frequency of the light produced and the accelaration of the charge ?

In general, the electromagnetic radiation from an accelerated charge does not have a single frequency but rather a distribution of frequencies. That distribution depends on the details of how the particle is accelerating.

The Liénard–Wiechert fields of an arbitrarily moving point charge is known. If you specify exactly how the charged particle is moving, you can use these fields to calculate the angular distribution and the frequency distribution of the radiated energy.

For example, for a particle of charge $$e$$ undergoing circular motion with uniform speed $$v$$ and circular radius $$\rho$$, the frequency distribution of the energy is calculated and graphed here. It is

$$\frac{dE}{d\omega}=\frac{\sqrt{3}e^2}{4\pi\epsilon_0c}\gamma\frac{\omega}{\omega_c}\int_{\omega/\omega_c}^\infty K_{5/3}(x)\,dx,$$

where the critical frequency $$\omega_c$$ (roughly the frequency at which the radiated energy is maximum) is

$$\omega_c=\frac{3}{2}\frac{c}{\rho}\gamma ^3.$$

Here $$\gamma=1/\sqrt{1-v^2/c^2}$$ is the usual Lorentz factor that depends on the speed of the particle.

For acceleration in a straight line, the frequency distribution is different. The details of the acceleration matter.

For more complicated trajectories, it may not be possible to get an analytic result, in which case numerical techniques would be necessary. But, in principle, everything about the radiation is calculable if one knows how the particle is moving.

No. Accelerated particle does not typically produce harmonic waves unless its motion is also harmonic (circular motion, or harmonically oscillating motion in a line). The radiation pattern depends on details of the motion. The pattern can be resolved via Fourier analysis into harmonic components, but in general there will contributions due to all frequencies. These contributions are not simple functions of particle state at a single time, but depend on the whole motion during the time interval in which the radiation was produced.