In a medium characterized by intrinsic impedance \(\eta=|\eta| e^{j \phi}\), a linearly polarized plane wave propagates, with magnetic field given as \(\mathbf{H}_{s}=\) \(\left(H_{0 y} \mathbf{a}_{y}+H_{0 z} \mathbf{a}_{z}\right) e^{-\alpha x} e^{-j \beta x} .\) Find \((a) \mathbf{E}_{s} ;(b) \mathcal{E}(x, t) ;(c) \mathcal{H}(x, t) ;(d)\langle\mathbf{S}\rangle .\)

Question: Given the plane wave's magnetic field component, $\mathbf{H_s} = \left(H_{0 y} \mathbf{a}_{y} + H_{0 z} \mathbf{a}_{z} \right) e^{-\alpha x} e^{-j \beta x}$, and intrinsic impedance $\eta$, find the corresponding electric field component, the instantaneous electric and magnetic fields, and the time-averaged Poynting vector. Answer: Electric field component: $\mathbf{E_s} = |\eta| e^{j \phi} \left(H_{0 y} \mathbf{a}_y e^{-\alpha x} e^{-j \beta x} + H_{0 z} \mathbf{a}_z e^{-\alpha x} e^{-j \beta x}\right)$ Instantaneous electric field: $\mathcal{E}(x, t) = |\eta| \left(H_{0 y} \mathbf{a}_y e^{-\alpha x} e^{-j (\beta x - \omega t)} + H_{0 z} \mathbf{a}_z e^{-\alpha x} e^{-j (\beta x - \omega t)}\right)$ Instantaneous magnetic field: $\mathcal{H}(x, t) = \left(H_{0 y} \mathbf{a}_{y}+H_{0 z} \mathbf{a}_{z}\right) e^{-\alpha x} e^{-j (\beta x - \omega t)}$ Time-averaged Poynting vector: $\langle \mathbf{S} \rangle = \frac{1}{2} \text{Re}\{(|\eta| e^{j \phi} \left(H_{0 y} \mathbf{a}_y e^{-\alpha x} e^{-j \beta x} + H_{0 z} \mathbf{a}_z e^{-\alpha x} e^{-j \beta x}\right)) \times (\left(H_{0 y} \mathbf{a}_{y}+H_{0 z} \mathbf{a}_{z}\right) e^{-\alpha x} e^{j \beta x})\}$