4. (25+5 points) As shown in the figure, the space is divided into two parts at $z=3$. The first part of the space ($z>3$) is filled with a dielectric material with a permittivity of $\epsilon_1$, and the second part ($z<3$) is filled with a dielectric with $\epsilon_2=5\epsilon_1$. The permeabilities of the two media are related as $\mu_2=4\mu_1$. It is given that there is no free charge density in the interface $\rho_s=0$. However, there is a surface current density $J_s\neq0$. The field intensities $E_1$, $H_1$, and $H_2$ are given as
$E_1(r)=[3yx\hat{x}-yz\hat{y}+3xz\hat{z}]$
$H_1(r)=[y^2\hat{x}-xy\hat{y}+3x\hat{z}]$
$H_2(r)=[3y^2\hat{x}-(z+x)\hat{y}+Ax\hat{z}]$
a. (5 points) Find the constant A.
b. (15 points) Find the electric field intensity $E_2(r)$ at $z=3^-$.
c. (10 points) Find the surface current density $J_s(r)$.
