Question 6

The force on a particle with mass $m$ and charge $q$ moving with velocity v in an electric field E and in a magnetic field B is

$${\bf F} = q[ {\bf E} + {\bf v} \times {\bf B} ].$$

Consider this motion from the viewpoint of a rotating frame of reference and show that, with an appropriate choice of angular velocity, the acceleration of the particle in this frame can be written in the form

$$m{\bf a}_r=q{\bf E}_{\rm eff}$$

where the ``effective electric field" in this frame ${\bf E}_{\rm eff}({\bf x})$ depends on the particle's position but not on its velocity.

According to classical physics an electron can move around a proton in an elliptical orbit. Describe the effect of a weak magnetic field on this orbit. [Assume that the field is weak enough that terms of order $B^2$ can be neglected.]

[Hints can be found here, but do not follow this link until you have attempted the question on your own.]