R3REL

1. Inverse Compton scattering: a low-energy photon of energy [math]\displaystyle{ \epsilon }[/math] is hit by a fast electron of energy E. Calculate the energy distribution of the final photon, and its frequency, as functions of the diffusion angle.
2. Can an electron be in uniform circular motion in a constant magnetic field [math]\displaystyle{ \overrightarrow B=(0,B,0) }[/math]? If no, explain what curve would correspond to the trajectory. If yes, calculate the angular frequency of the motion.
3. Calculate the worldline [math]\displaystyle{ (x(t),y(t),z(t)) }[/math] of an electron initially at rest, and in a fixed electromagnetic field [math]\displaystyle{ \overrightarrow E=(E,0,0) }[/math] and [math]\displaystyle{ \overrightarrow B=(0,B,0) }[/math].
4. Write the current [math]\displaystyle{ J^\mu }[/math] and the stress-energy tensor [math]\displaystyle{ T^{\mu\nu} }[/math] of an electron of charge [math]\displaystyle{ e }[/math] and rest mass [math]\displaystyle{ m_0 }[/math] on the worldline [math]\displaystyle{ (t,\overrightarrow x(t)) }[/math]. The proper time can be calculates from [math]\displaystyle{ c^2d\tau^2=c^2dt^2-d\overrightarrow x\cdot d\overrightarrow x }[/math]. What happens if one chooses the negave solution for [math]\displaystyle{ d\tau }[/math]? What would be the mass and charge of the associated particle?
5. Show that [math]\displaystyle{ \overrightarrow E }[/math] and [math]\displaystyle{ \overrightarrow B }[/math] transform as 3-vectors under rotations.