Hint 6

Write down the equation of motion from the viewpoint of a frame rotating with angular velocity $\omega$, including the centrifugal and Coriolis terms. Remember to express the inertial-frame velocity appearing the magnetic part of the force in terms of the velocity in the rotating frame. You should find that if you choose

$$\mbox{\boldmath $\omega$}=-{q\over 2m}{\bf B},$$

one of the inertial terms exactly cancels the force due to the magnetic field. The resulting equation of motion should then involve an apparent force that depends on the particle's position, but not on its velocity:

$$m{\bf a}_r=q{\bf E}+m\mbox{\boldmath $\omega$}\times (\mbox{\boldmath $\omega$}\times{\bf x}).$$

In a weak magnetic field, you may neglect the final term of order $B^2$, in which case the apparent force is just equal to the original electric force $q{\bf E}$. The electric field around the proton is given by the usual inverse-square law. Classically, an electron follows an elliptical orbit in such a field. (This is just like the corresponding planetary orbits to be discussed in the Gravitation section of the course.) Now describe how this motion looks from the standpoint of a nonrotating observer.

This effect is known as Larmor precession. Its quantum mechanical analogue is important in understanding the spectra of atoms in magnetic fields.