Question #43633

In a quantum-classical hybrid experiment, an electron (charge $e$) is accelerated from rest through the instantaneous potential difference across an inductor (inductance $L$) in a series $RL$ circuit. The circuit is connected to a DC battery (EMF $V$) and a resistor (resistance $R$). The switch is closed at $t=0$. The instantaneous kinetic energy $E(t)$ acquired by the electron is determined by the induced EMF across the inductor at that instant. Due to the rapid transient decay of the magnetic field, the energy of the electron has an uncertainty $\Delta E$ coupled to the uncertainty in its emission time $\Delta t$. Assuming the energy uncertainty is given by the first-order variation $\Delta E = \left| \frac{dE}{dt} \right|_{t=0} \Delta t$, find the minimum uncertainty in the emission time $\Delta t$ according to the Heisenberg uncertainty principle $\Delta E \Delta t \ge \frac{\hbar}{2}$. (All quantities are in SI units, and $\hbar$ is the reduced Planck constant).