Question #42647

Consider a cylindrical plasma column of radius $R$ with its axis aligned along the $x$-axis. The equilibrium electron number density varies radially as $n_0(r) = \eta r$, where $r$ is the distance from the $x$-axis and $\eta$ is a positive constant. An electron plasma wave propagates in the positive $x$-direction. Assume the electron number density remains approximately equal to its equilibrium value $n_0(r)$ for calculating the current density. At $t=0$, the spatial profile of the electron drift velocity is given by $\vec{v}_d(x,0) = v_0 \sin(kx) \hat{i}$ (where $v_0 > 0$ and $k > 0$). The wave has a temporal angular frequency $\omega$. What is the net charge $\Delta Q$ that flows through the cross-section at $x=0$ in the positive $x$-direction during the time interval $t=0$ to $t=\frac{\pi}{\omega}$? (Here $e$ represents the elementary positive charge).