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!
