Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 4 / 10A (Dynamics and Relativity)

The Lorentz force and Newton's second law giveWriting $\Omega =qB/m$, the component equations areThe initial velocity therefore givesand integration with the initial position givesThis is helical motion in a uniform magnetic field: a helix of radius $v/\Omega$ around the line $x=1+v/\Omega$, $y=0$, with its axis parallel to the magnetic field.

The three fixed charges are related by rotations through $2\pi /3$ about the origin. Their three Coulomb's law forces on a charge placed at the origin have equal magnitudes and directions separated by $2\pi /3$, so their vector sum vanishes. Thusis an equilibrium point.

Let $\rho =a/\sqrt{3}$ be the distance from the origin to each fixed charge, and denote their position vectors by ${\mathbf{R}}_j$. For a small planar displacement $\mathbf{r}$, the Taylor expansioncan be summed using the threefold symmetry relationsThe electric potential due to the fixed charges is consequentlyThe moving charge has electrostatic potential energy of point charges $U=q\Phi$. Because the quadratic coefficient is positive, the origin is a strict local minimum of $U$, and the equilibrium is stable.

Taking the gradient of the quadratic potential energy found in part (ii) resolves the linearized force into identical $x$ and $y$ components:Each component therefore performs simple harmonic motion withThus the angular frequency is