22-Elec-A7 Electromagnetics · May 2013

Question 5 of 8: Surface Current Induced by Oblique Incidence on a Conducting Plane

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, May 2013 — 07-Elec-A7, Electromagnetics. Three hours, closed book; one of two approved calculators (Casio or Sharp) permitted. Eight questions, all of equal value; any five constitute a complete paper and only the first five presented are marked. All eight are solved here, since the set is a study resource. Aids printed on the cover page: ε₀ = 8.85 × 10⁻¹² F/m and μ₀ = 4π × 10⁻⁷ H/m; Question 2 additionally supplies the quarter-wave relation.

Reference texts (22-Elec-A7 Electromagnetics).

D. M. Pozar, Microwave Engineering, 4th ed. — transmission lines, matching, waveguides.
M. N. O. Sadiku, Elements of Electromagnetics, 7th ed. — plane waves, guided waves, radiation.
W. H. Hayt and J. A. Buck, Engineering Electromagnetics, 9th ed. — field theory and transients on lines.
C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. — short elements, image theory.
F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 8th ed. — oblique incidence, surface currents.
S. J. Chapman, Electric Machinery Fundamentals, 5th ed. — rotating-loop induction and torque (Question 3).

Conventions used throughout. The paper itself quotes propagation velocities as $3\times10^{8}\ \text{m/s}$, so free space is taken as $c=3.00\times10^{8}\ \text{m/s}$ and the intrinsic impedance as $\eta_0=\mu_0 c=120\pi=376.99\ \Omega$. Lines are treated as lossless, and "rms" is stated explicitly wherever the exam asks for it.

Question 5: Surface Current Induced by Oblique Incidence on a Conducting Plane (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. A horizontally travelling 10 GHz plane wave with a vertical electric field and 1 W/m² of power density strikes a vertical perfectly conducting plane at 45° from the normal.

Given data
Quantity	Symbol	Value
Frequency	$f$	10.0 GHz ($\lambda=30.0$ mm)
Incident power density (time average)	$S$	1.00 W/m²
Angle of incidence (from the normal)	$\theta_i$	45°
Polarization	—	vertical $\mathbf{E}$; propagation horizontal
Reflector	—	vertical perfectly conducting plane
Intrinsic impedance of free space	$\eta_0$	$120\pi=377\ \Omega$

Find. (i) the rms magnitude of the induced surface current density and how it is distributed over the plane, and (ii) the direction in which that current flows.

Figure 5.1 — Plan view. The plane of incidence is horizontal, so the vertical electric field is perpendicular to it and lies wholly in the reflecting surface; the induced current points out of the page, i.e. vertically up the wall.

Approach. Convert the power density to rms field amplitudes, classify the polarization from the geometry, use the perfect-conductor boundary condition that the tangential magnetic field doubles, and evaluate $\mathbf{J}_s=\hat{\mathbf{n}}\times\mathbf{H}$.

Recover the incident field amplitudes. For a plane wave in free space $S=E_{rms}^{2}/\eta_0$, hence $$E_{rms}=\sqrt{S\eta_0}=\sqrt{1.00\times376.99}=19.42\ \text{V/m},\qquad H_{rms}=\frac{E_{rms}}{\eta_0}=\sqrt{\frac{S}{\eta_0}}=51.50\ \text{mA/m}$$
Classify the polarization from the geometry. Both the propagation direction and the surface normal are horizontal, so the plane of incidence is horizontal. The electric field is vertical, therefore perpendicular to that plane and entirely tangential to the wall: this is perpendicular (TE, or horizontal-polarization-relative-to-the-plane-of-incidence) reflection, for which a perfect conductor gives $\Gamma_{\perp}=-1$. The magnetic field lies in the horizontal plane of incidence, perpendicular to the ray.
Apply the perfect-conductor boundary condition. With $\Gamma=-1$ the total tangential electric field vanishes, while the tangential magnetic fields of the incident and reflected waves add in phase, doubling. The component of $\mathbf{H}_i$ that is tangential to the wall is reduced by the obliquity: $$H_{tan,i}=H_{rms}\cos\theta_i=51.50\times\cos45^\circ=36.42\ \text{mA/m}$$ so the total tangential field at the surface is $2\times36.42=72.84$ mA/m.
Evaluate the surface current density. The boundary condition on a perfect conductor is $\mathbf{J}_s=\hat{\mathbf{n}}\times\mathbf{H}_{tot}$, giving $$J_{s,rms}=2H_{rms}\cos\theta_i=2\times51.50\times0.7071=\boxed{72.8\ \text{mA/m (rms)}}$$
Establish the direction. The outward normal $\hat{\mathbf{n}}$ is horizontal and the total tangential $\mathbf{H}$ is also horizontal and perpendicular to it, so their cross product is vertical. Taking the wall as the $y$–$z$ plane with $z$ up and the wave arriving in the $x$–$y$ plane, $\hat{\mathbf{n}}\times\mathbf{H}_{tot}$ points along $+\hat{\mathbf{z}}$: $$\boxed{\mathbf{J}_s\ \text{is vertical, parallel to the incident }\mathbf{E}}$$ The current therefore runs up and down the wall. This is the expected result, because the current must follow the tangential electric field that drives it, and that field is vertical.
Describe the pattern over the surface. The phase of the incident wave advances along the wall as $e^{-j\beta y\sin\theta_i}$, so the current is a travelling wave of uniform magnitude whose phase repeats over $$\lambda_{s}=\frac{\lambda}{\sin\theta_i}=\frac{30.0\ \text{mm}}{0.7071}=42.4\ \text{mm}=4.24\ \text{cm}$$ measured horizontally along the wall in the direction of the ray's horizontal projection. There is no standing-wave variation across the surface; the standing wave produced by the reflection exists in front of the wall, along the normal.

A useful sanity check is the limiting behaviour. At normal incidence ($\theta_i=0$) the expression gives $J_s=2H=103$ mA/m, the familiar doubling result; at grazing incidence it falls to zero, because a wave skimming the surface presents almost no tangential magnetic field. The 45° case sits between them at $\sqrt2\,H$.

Final results — Question 5
Quantity	Value
Incident rms electric field	19.4 V/m
Incident rms magnetic field	51.5 mA/m
Polarization relative to the plane of incidence	perpendicular (TE), $\Gamma_{\perp}=-1$
(i) RMS surface current density	72.8 mA/m, uniform in magnitude
Spatial period of the pattern along the wall	4.24 cm ($\lambda/\sin\theta_i$)
(ii) Direction of the surface current	vertical, parallel to the incident $\mathbf{E}$

Topic. Oblique incidence of a plane wave on a perfect conductor, and the surface current it induces.

Key relations. $S=E_{rms}^{2}/\eta_0$; $\eta_0=120\pi$; $\Gamma_{\perp}=-1$ at a perfect conductor for tangential $\mathbf{E}$; $\mathbf{J}_s=\hat{\mathbf{n}}\times\mathbf{H}_{tot}$ with $H_{tot,tan}=2H_i\cos\theta_i$; surface phase period $\lambda/\sin\theta_i$.

Why this works. A perfect conductor cannot support a tangential electric field, so the reflected wave must arrive with exactly the amplitude and sign that cancels the incident tangential $\mathbf{E}$. The same superposition that cancels the electric field reinforces the tangential magnetic field, and Ampere's law at the boundary converts that doubled magnetic field into a physical sheet current. The $\cos\theta_i$ factor appears because only the projection of $\mathbf{H}$ onto the surface participates: at oblique incidence part of the magnetic field points along the normal and contributes nothing. Finally, the phase of the incident wave varies along the surface at the rate $\beta\sin\theta_i$, so the induced current is a travelling wave whose spatial period stretches from $\lambda$ at grazing incidence to infinity at normal incidence.

Common pitfall. Omitting the $\cos\theta_i$ obliquity factor and reporting 103 mA/m, which is the normal-incidence answer. A second common error is to look for a standing-wave pattern of current magnitude on the wall: with a single incident plane wave the magnitude is uniform, and only the phase varies along the surface.

Source. Ulaby & Ravaioli, Fundamentals of Applied Electromagnetics, 8th ed., Ch. 8 (oblique incidence); Sadiku, Elements of Electromagnetics, 7th ed., Ch. 10 (plane waves at boundaries).

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