Past exam of the physics 2 paper of the University of Oxford / 2013 resit / 1

The magnetic Lorentz force isSince the given velocity is perpendicular to the magnetic flux density, its magnitude is $|q|vB$. Its direction follows the vector product for positive charge and reverses for negative charge.

The instantaneous power delivered by the magnetic field isTherefore $\boxed{W=0}$ over any part of the orbit. The magnetic force changes direction of motion but leaves the speed and kinetic energy constant.

Let ${\mathbf{r}}_P$ be the initial position, $\hat{\mathbf{b}}=\mathbf{B}/B$, and $\Omega =qB/m$, including the sign of $q$. Solving $\dot{\mathbf{v}}=\Omega \mathbf{v}\times \hat{\mathbf{b}}$ givesThusThe locus is a circle in the plane perpendicular to $\mathbf{B}$, centred at ${\mathbf{r}}_c={\mathbf{r}}_P+({\mathbf{v}}_0\times \hat{\mathbf{b}})/\Omega$, withThis also follows by equating the magnetic force to the centripetal force. The nonrelativistic angular frequency is $|q|B/m$.