A A \(2 \mathrm{GHz}\) uniform plane wave has an amplitude \(E_{y 0}=1.4 \mathrm{kV} / \mathrm{m}\) at \((0,0,0,\), \(t=0\) ) and is propagating in the \(\mathbf{a}_{z}\) direction in a medium where \(\epsilon^{\prime \prime}=1.6 \times\) \(10^{-11} \mathrm{~F} / \mathrm{m}, \epsilon^{\prime}=3.0 \times 10^{-11} \mathrm{~F} / \mathrm{m}\), and \(\mu=2.5 \mu \mathrm{H} / \mathrm{m}\). Find \((\) a \() E_{y}\) at \(P(0,0,1.8 \mathrm{~cm})\) at \(0.2 \mathrm{~ns} ;(b) H_{x}\) at \(P\) at \(0.2 \mathrm{~ns}\).

We will first find the attenuation constant, phase constant, impedance, and phase velocity of the wave using the given values of ε', ε'' and µ. Let's start by finding the complex permittivity ε*, complex propagation constant γ, and intrinsic impedance η. \(\epsilon^{*} = \epsilon' - j\epsilon'' = 3.0 \times 10^{-11} - j1.6 \times 10^{-11}\) F/m We also know the angular frequency, ω, and should calculate it: \(\omega = 2\pi f = 2\pi(2 \times 10^9)\) rad/s Now, let's compute the complex propagation constant, γ, using: \(\gamma = j\omega\sqrt{(\mu\epsilon^*)} = j(2\pi \times 2 \times 10^9)\sqrt{(2.5 \times 10^{-6})(3.0 \times 10^{-11} - j1.6 \times 10^{-11})}\) After calculating the complex propagation constant, we can find the attenuation constant, α, and phase constant, β, using: \(\alpha = \operatorname{Re}(\gamma)\) and \(\beta = \operatorname{Im}(\gamma)\) Next, let's find the intrinsic impedance, η, using: \(\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon'}}\) Now that we have α, β, and η, we can proceed to phase 2.

Using the values calculated in Phase 1, we can find Ey and Hx at the given point (0,0,1.8 cm) and time (0.2 ns). (a) Ey at point P and time t can be calculated using: \(E_y = E_{y0} e^{(-\alpha z)}\cos{(\omega t - \beta z)}\) Substitute the known values: \(E_{y0} = 1.4 \times 10^3\) V/m, z = 1.8 × 10⁻² m, and t = 0.2 × 10⁻⁹ s, and use α, β, and ω derived in Phase 1. \(E_{y} = (1.4 \times 10^3) e^{(-\alpha \times 1.8 \times 10^{-2})}\cos{((2\pi \times 2 \times 10^9)(0.2 \times 10^{-9}) - \beta \times 1.8 \times 10^{-2})}\) Solve for Ey. (b) Now, let's find the magnetic field component Hx at point P at time 0.2 ns. Using the given values and the values derived in Phase 1, Hx can be calculated using: \(H_x(z, t) = \frac{E_y(z, t)}{\eta}\) Substitute the derived value of Ey and η into the equation: \(H_{x}(1.8 \times 10^{-2}, 0.2 \times 10^{-9}) = \frac{E_{y}(1.8 \times 10^{-2}, 0.2 \times 10^{-9})}{\eta}\) Solve for Hx. By following these calculations step-by-step, you can find the values of Ey and Hx at the specified point and time.