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# 4. ( 35 points) Electromagnetic waves. (This problem is similar to the example with two parallel semi-infinite plates.) A semi-infinite single conducting plate is kept at an alternating voltage $V(t)$ which results in a current density $J(x, t)(A m p . / m)$, and a charge density $\sigma(x, t)\left(C / m^{2}\right)$. a) ( 5 points) What is the direction and magnitude of the electric field $E$, on both sides of the plate, at distance $x$ down the plate? Do not derive the result from Gauss' Law, just quote it. b) ( 5 points) What is the direction and magnitude of the magnetic field $B$, on both sides of the plate, at distance $x$ down the plate? Use Ampere's Law around the rectangle $R_{1}$. c) ( 5 points) Use Faraday's Law of induction with the rectangle $R_{2}$ to derive the relation between $\frac{\partial E_{\mathrm{z}}}{\partial x}$ and $\frac{\partial B_{y}}{\partial t}$. d) ( 5 points) Use the displacement current term in the Ampere-Maxwell Equation with the rectangle $R_{3}$ to derive the relation between $\frac{\partial B_{y}}{\partial x}$ and $\frac{\partial E_{z}}{\partial t}$. e) ( 5 points) Obtain the wave equation for $B_{y}$ from your results for c) and d). f) ( 5 points) In your result for d), substitute for $B_{y}$ and $E_{z}$ in terms of $J(x, t)$ and $\sigma(x, t)$ using your results from b) and a). You will obtain a relation between between $\frac{\partial J}{\partial x}$ and $\frac{\partial \sigma}{\partial t}$. g) ( 5 points) In your result for $\frac{\partial J}{\partial x}$ note that $d J=J(x+d x)-J(x)$. Now multiply both sides of your result by $H d x$ referring to the rectangle $R_{4}$. What is the meaning of your result $g$ )?  