References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 11, Post 16.
One common instance of an accelerated charge is a charge moving in a circle. In this case the particle’s instantaneous velocity is always perpendicular to its instantaneous acceleration . This is known as synchrotron radiation, since it is the radiation given off by particles in a synchrotron particle accelerator, where charged particles move in circular orbits between the poles of a magnet.
We can use the Liénard formula to work out the power radiated by such a charge:
At one instant of time, we can take
where we’re using our usual shorthand for trig functions: , and so on. We can now work out the components of 1:
Taking the square of this last vector leads to a lengthy expression which can be simplified by applying repeatedly. We get, using :
We can simplify this as follows. The first and seventh terms combine to give
Combining this with the eighth term:
Combining this with the fourth and last terms we get
The second, third and sixth terms combine to give
Finally, the fifth and ninth terms cancel, so we’re left with
Putting everything together we get
To get the total power, we need to integrate this over all solid angles, so we get
The integral over is easy, using
so we’re left with the integral over :
This nasty looking integral can be done by using partial fractions, since it is the ratio of two polynomials in . I did the integral using Maple, but if you’re interested in doing it by hand, the partial fraction decomposition is
The presence of the extra from the solid angle element saves the day, since it multiplies each term in the partial fraction expansion, providing the derivative of on the top of each fraction. For example
with the other two terms having similar integrals.
The result of the integral is
so we get for the total power