# Special Relativity for Synchrotron Radiation

For the last few hours, I tried to solve an exercise about synchrotron radiation but can't get to a solution. I think there are some concepts about special relativity that I didn't understand during the lecture. The exercise is the following:

It is given that the power P radiated by an accelerated charge e is: $$P = \frac{2}{3} \hbar \alpha \gamma^{6}(\dot{\vec{\beta}}^2-(\vec{\beta} \times \dot{\vec{\beta}})^2)$$ I am asked to express this in terms of the bending radius $$\rho$$ and the particle energy. I can assume highly relativistic particles.

It is clear to me that the cross product is a simple multiplication since the velocity and acceleration is perpendicular. But how can I express $$\beta$$ in a way that the term only depends on the particle energy?

I started with $$\vec{\beta} = \frac{\vec{p}c}{E}$$ and with that $$\dot{\vec{\beta}} = \frac{\dot{\vec{p}}c}{E}$$

Now $$\dot{p}$$ has to be the force (Lorentz force) so $$\dot{p} = qvB$$ and the magnetic field can be determined by the centripetal force: $$qvB = \frac{mv^2}{\rho}$$ $$\Leftrightarrow$$ $$B = \frac{mv}{q\rho}$$

$$\rightarrow \quad$$ $$\dot{\vec{p}} = \frac{mv^2}{\rho}$$

And now I'm stuck. Because if I continue like that it will end up with an expression dependent on the velocity, mass, energy of the particle and bending radius. And without a given mass I can't determine the radiation power P.

I think I made a mistake when saying $$\dot{\vec{p}} = Force$$ because I didn't consider special relativity effects at all but I have no idea what to use instead.

Thank you very much for your help!

• Oh well. If your reasoning leads to a result that you doubt, do not just stop, but check your reasoning and write down its unescapable result. If the reasoning checks out, the result is a correct conclusion from the assumptions and the doubt is ungrounded. In this case, if $P$ turns out to be a function of both energy and mass, that is a fine result. The Larmor power for ultrarelativistic particle is a function of Lorentz factor and radius, and the Lorentz factor is not determined solely by energy. One has to know the mass. – Ján Lalinský Feb 5 '19 at 0:21

$$\def\vb{\vec\beta}$$ First hint. You noted that $$\vb$$ and $$\dot\vb$$ are orthogonal. Then the expression in parentheses is $$\Bigl|\dot\vb\Bigr|^2\,(1 - \beta^2).$$
Second hint. For circular motion acceleration = $$v^2/\rho$$ so $$\Bigl|\dot\vb\Bigr| = {c\,\beta^2 \over \rho} \simeq {c \over \rho}.$$
And without a given mass I can't determine the radiation power $$P$$.
Indeed you can't. You have a $$\gamma$$ which must be converted in energy and mass is required.