- #1

- Homework Statement
- A particle with charge q and a relativistic velocity $$\vec{v}_0$$ penetrates a medium in which it is decelerated by a force proportional to its velocity, $$\vec{F} = -\alpha\vec{v}$$, with $$\alpha$$ constant. How much energy is emitted in the form of electromagnetic radiation until the particle stops? Assume the medium behaves like vacuum for the radiation.

- Relevant Equations
- relativistic Larmor formula: $$P = \frac{2e^2\gamma^6}{3c}[\dot{\beta}^2 - (\vec{\beta}\times\dot{\vec{\beta}})^2]$$ where $$\vec{\beta} = \vec{v}/c$$ and $$\gamma$$ is the usual Lorentz factor.

Honestly, folks, I don't even know how to start. I included in the Relevant Equations section the relativistic generalization of the Larmor formula according to Jackson, because that's the equation for the power emitted by an accelerated particle, but I don't see how that gets me very far.

The only thing I realized while thinking about this problem is that the second term inside the brackets in the equation, $$(\vec{\beta}\times\dot{\vec{\beta}})^2$$, is zero, because the velocity and the acceleration are collinear.

Any help or insight would be most appreciated!

The only thing I realized while thinking about this problem is that the second term inside the brackets in the equation, $$(\vec{\beta}\times\dot{\vec{\beta}})^2$$, is zero, because the velocity and the acceleration are collinear.

Any help or insight would be most appreciated!