Question 8.7: Region 0 ≤ z ≤ 2 m is occupied by an infinite slab of permea...

(a) By definition

J=\nabla\times H=\nabla\times\frac{B}{\mu_{o}\mu_{r}}=\frac{1}{4\pi\times10^{-7}(2.5)}\left(\frac{\partial B_{y}}{\partial x}-\frac{\partial B_{x}}{\partial y}\right)a_{z}

=\frac{10^{6}}{\pi}(-5-10)10^{-3}a_{z}=-4.775a_{z}kA/m^{2}

(b)

J_{b}=\chi_{m}J=(\mu_{r}-1)J=1.5(-4.775a_{z})\cdot 10^{3}=-7.163a_{z}kA/m^{2}

(c)

M=\chi_{m}H=\chi_{m}\frac{B}{\mu_{o}\mu_{r}}=\frac{1.5(10ya_{x}-5xa_{y})\cdot10^{-3}}{4\pi\times10^{-7}(2.5)}=4.775ya_{x}-2.387xa_{y}kA/m

(d)

K_{b}=M\times a_{n^{\cdot}}

Since z=0 is the lower side of the slab occupying 0\leq z\leq 2, a_{n}=-a_{z^{\cdot}}. Hence

K_{b}=(4.775ya_{x}-2.387xa_{y})\times(-a_{z})=2.387xa_{x}+4.775ya_{y}kA/m