Question 1 of 10: Oblique Incidence at a Free-Space–Silica Interface

Question 1: Oblique Incidence at a Free-Space–Silica Interface (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Interface at $z=0$: free space ($\varepsilon_{r1}=1$) for $z<0$, silica ($\varepsilon_{r2}=3.8$, lossless) for $z>0$; both non-magnetic. Plane wave at $f=1$ GHz (not needed below — none of the requested quantities depend on frequency), $\theta_i=30^\circ$, $\mathbf E^i$ polarized along $\hat y$ (out of the $xz$-plane of incidence, i.e. perpendicular/TE polarization), $|E^i|=2$ V/m.

Find. $\theta_t$; the perpendicular-polarization Fresnel coefficients $\Gamma_\perp,\tau_\perp$; and the magnetic field strengths $H^i$ (in $z<0$) and $H^t$ (in $z>0$).

Approach. Snell's law fixes $\theta_t$ from the two permittivities; the perpendicular-polarization boundary conditions (continuity of $E_y$ and $H_x$) give $\Gamma_\perp,\tau_\perp$ in terms of the intrinsic impedances and $\theta_i,\theta_t$; the magnetic field of a uniform plane wave is always $|H|=|E|/\eta$ in its own medium, so $H^i$ follows directly from $E^i$ and $H^t$ follows from $E^t=\tau_\perp E^i$.

1. Intrinsic impedances. $$\eta_1=\frac{\eta_0}{\sqrt{\varepsilon_{r1}}}=376.7\ \Omega,\qquad \eta_2=\frac{\eta_0}{\sqrt{\varepsilon_{r2}}}=\frac{376.7}{\sqrt{3.8}}=193.3\ \Omega.$$
2. Part (b) — Angle of refraction (Snell's law). $$\sqrt{\varepsilon_{r1}}\sin\theta_i=\sqrt{\varepsilon_{r2}}\sin\theta_t \Rightarrow \sin\theta_t=\sin30^\circ\sqrt{\frac{1}{3.8}}=0.2565 \Rightarrow \boxed{\theta_t=14.86^\circ}.$$
3. Part (c) — Perpendicular (TE) Fresnel coefficients. With $\cos\theta_i=0.8660$, $\cos\theta_t=0.9665$: $$\Gamma_\perp=\frac{\eta_2\cos\theta_i-\eta_1\cos\theta_t}{\eta_2\cos\theta_i+\eta_1\cos\theta_t}=\frac{(193.3)(0.8660)-(376.7)(0.9665)}{(193.3)(0.8660)+(376.7)(0.9665)}=\boxed{-0.370}$$ $$\tau_\perp=1+\Gamma_\perp=\boxed{0.630}.$$ Check: continuity of $E_y$ at $z=0$ requires $1+\Gamma_\perp=\tau_\perp$ exactly — satisfied.
4. Part (a)(i) — Magnetic field strength in $z<0$. A uniform plane wave obeys $|H|=|E|/\eta$ in its own medium regardless of propagation angle (the impedance relates $E$ and $H$ magnitudes; only their vector directions depend on the wave's direction of travel). In $z<0$ the incident wave carries $$H^i=\frac{E^i}{\eta_1}=\frac{2}{376.7}=\boxed{5.31\times10^{-3}\ \text{A/m} = 5.31\ \text{mA/m}}.$$
5. Part (a)(ii) — Magnetic field strength in $z>0$. In $z>0$ only the transmitted wave exists, with $E^t=\tau_\perp E^i=(0.630)(2)=1.260$ V/m, so $$H^t=\frac{E^t}{\eta_2}=\frac{1.260}{193.3}=\boxed{6.52\times10^{-3}\ \text{A/m}=6.52\ \text{mA/m}}.$$