Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 3C (Dynamics and Relativity)

Take the scalar product of the Lorentz force equation with the constant magnetic field $\mathbf{B}$. Since $\mathbf{E}\cdot \mathbf{B}=0$ and $(\dot{\mathbf{x}}\times \mathbf{B})\cdot \mathbf{B}=0$,The initial velocity also obeys $\mathbf{v}\cdot \mathbf{B}=0$, so $\mathbf{x}\cdot \mathbf{B}$ remains equal to ${\mathbf{x}}_0\cdot \mathbf{B}$. The trajectory therefore lies in the planethe plane through ${\mathbf{x}}_0$ perpendicular to $\mathbf{B}$.

Let $\mathbf{E}=E\hat{\mathbf{x}}$ and $\mathbf{B}=B\hat{\mathbf{y}}$, and choose $\hat{\mathbf{z}}=\hat{\mathbf{x}}\times \hat{\mathbf{y}}$. Part (a) gives $y=0$. With the origin at ${\mathbf{x}}_0$, the remaining Cartesian components areThe initial conditions are $x=z=\dot{x}=\dot{z}=0$. Integrating the second equation gives $\dot{z}=\omega x$, soSolving this forced harmonic oscillator and then integrating $\dot{z}=\omega x$ yieldsThus the position vector is ${\mathbf{x}}_0+x(t)\hat{\mathbf{x}}+z(t)\hat{\mathbf{z}}$. The oscillation occurs at the signed cyclotron frequency, superposed on the usual E-cross-B drift.