Question #43609

A large metallic plate lying in the $x$-$y$ plane ($z=0$) has a work function $W$. It is illuminated by monochromatic light of photon energy $E_{ph}$ (where $E_{ph} > W$). Consider a photoelectron emitted from the origin with the maximum possible kinetic energy, at an angle $\theta$ with the $+z$-axis. In the region $z > 0$, a uniform electric field $\vec{E} = E_0 \hat{k}$ is present, which acts as a retarding field for the emitted electrons. Let $z_0$ be the maximum $z$-coordinate reached by this specific electron. Suppose that exactly at the plane $z = z_0$, the electric field abruptly terminates, and a non-uniform magnetic field $\vec{B} = B_0(1 + \alpha z)\hat{k}$ exists for all $z \ge z_0$. What is the radius of curvature of the electron's trajectory immediately after it enters the magnetic field region? (Assume $m$ and $-e$ are the mass and charge of the electron, respectively. Neglect gravity. Let $\Delta E = E_{ph} - W$. $E_0, B_0$, and $\alpha$ are positive constants with appropriate SI units).