Question #82260

A parallel plate capacitor consists of two rectangular plates of length $L = 0.2 \text{ m}$ (along the $x$-axis) and width $W = 0.1 \text{ m}$ (along the $y$-axis), separated by a distance $d = 5 \text{ mm}$ along the $z$-axis. A constant potential difference $V_0 = 50 \text{ V}$ is maintained across the plates. An ideal, incompressible dielectric fluid flows between the plates with a uniform velocity $\vec{v} = 2.0 \hat{i} \text{ m s}^{-1}$. The permittivity of any individual fluid parcel remains constant as it moves (i.e., the material derivative $\frac{D\varepsilon}{Dt} = 0$). At a given instant $t=0$, the permittivity of the fluid between the plates varies continuously along the $x$-axis from $\varepsilon_1 = 3\varepsilon_0$ at the entrance ($x=0$) to $\varepsilon_2 = 8\varepsilon_0$ at the exit ($x=L$), where $\varepsilon_0$ is the vacuum permittivity. A uniform magnetic field $\vec{B} = 0.5 \hat{i} \text{ T}$ is also present. What is the magnitude of the total displacement current through the capacitor at $t=0$?