Engineering Physics

Chapter 6
Electromagnetic Waves, Transmission and Radiation

Bachelors Program in Electrical Engineering
Engineering Physics • Bachelor

Electromagnetic Waves, Transmission and Radiation

Introduction

Electromagnetic waves transport energy and information through free space, material media, transmission lines, and waveguides. Their behavior underlies radio communication, radar, satellite links, optical systems, microwave circuits, power-electronic equipment, and electromagnetic compatibility (EMC).

This chapter develops electromagnetic-wave behavior from Maxwell’s equations and connects field theory to practical engineering models. It distinguishes unbounded plane waves from guided waves, explains reflection and refraction at interfaces, and examines attenuation, skin depth, transmission-line mismatch, waveguide modes, antenna radiation, spectrum use, and interference control.

Learning Objectives

After completing this chapter, a student should be able to:

derive the electromagnetic wave equation from Maxwell’s equations;
determine wave velocity, wavelength, intrinsic impedance, power flow, attenuation, and skin depth;
analyze reflection and transmission at material interfaces;
apply transmission-line equations, reflection coefficients, and standing-wave ratio;
calculate rectangular-waveguide cutoff and propagation parameters;
interpret basic antenna quantities and perform a free-space link calculation;
identify conducted and radiated EMC mechanisms;
evaluate practical spectrum, safety, and regulatory constraints.

1. Maxwell’s Equations and Wave Propagation

For a linear, homogeneous, isotropic medium, the constitutive relations are \mathbf{D}=\varepsilon\mathbf{E}, \mathbf{B}=\mu\mathbf{H}, and \mathbf{J}=\sigma\mathbf{E}, where:

\mathbf{E} is electric-field intensity, V/m;
\mathbf{H} is magnetic-field intensity, A/m;
\mathbf{D} is electric flux density, C/m²;
\mathbf{B} is magnetic flux density, T;
\varepsilon is permittivity, F/m;
\mu is permeability, H/m;
\sigma is conductivity, S/m.

Maxwell’s differential equations are

\nabla\cdot\mathbf{D}=\rho_v

\nabla\cdot\mathbf{B}=0

\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}

\nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}

Here \rho_v is free volume charge density in C/m³. The term \partial\mathbf{D}/\partial t is displacement-current density and permits electromagnetic propagation even where conduction current is absent.

1.1 Derivation of the wave equation

Consider a source-free, homogeneous region. Taking the curl of Faraday’s law gives

\nabla\times(\nabla\times\mathbf{E})=-\mu\frac{\partial}{\partial t}(\nabla\times\mathbf{H})

Using \nabla\times(\nabla\times\mathbf{E})=\nabla(\nabla\cdot\mathbf{E})-\nabla²\mathbf{E}, with \nabla\cdot\mathbf{E}=0 in a homogeneous source-free region, produces

\nabla²\mathbf{E}=\mu\sigma\frac{\partial\mathbf{E}}{\partial t}+\mu\varepsilon\frac{\partial²\mathbf{E}}{\partial t²}

Similarly,

\nabla²\mathbf{H}=\mu\sigma\frac{\partial\mathbf{H}}{\partial t}+\mu\varepsilon\frac{\partial²\mathbf{H}}{\partial t²}

The first right-hand term represents conductive loss; the second supports wave propagation through energy exchange between electric and magnetic fields.

1.2 Uniform plane waves

Using the phasor convention e^{j\omega t}, a wave traveling in the positive z direction has the form

\mathbf{E}(z)=\mathbf{E}₀e^{-\gamma z}, \qquad \gamma=\alpha+j\beta

where \gamma is the propagation constant, \alpha is attenuation constant in Np/m, and \beta is phase constant in rad/m. In a general conducting medium,

\gamma=\sqrt{j\omega\mu(\sigma+j\omega\varepsilon)}

The intrinsic impedance is

\eta=\frac{E}{H}=\sqrt{\frac{j\omega\mu}{\sigma+j\omega\varepsilon}}

These expressions assume a linear, homogeneous, isotropic medium and sinusoidal steady state.

For a lossless medium, \sigma=0, so

\alpha=0, \qquad \beta=\omega\sqrt{\mu\varepsilon}, \qquad v_p=\frac{1}{\sqrt{\mu\varepsilon}}

\lambda=\frac{2\pi}{\beta}=\frac{v_p}{f}, \qquad \eta=\sqrt{\frac{\mu}{\varepsilon}}

In free space, v_p=c\approx2.998\times10⁸\ \text{m/s} and \eta₀\approx376.73\ \Omega. The vectors \mathbf{E}, \mathbf{H}, and the direction of propagation are mutually perpendicular.

1.3 Energy transport

The instantaneous power-flow density is represented by the Poynting vector:

\mathbf{S}=\mathbf{E}\times\mathbf{H} \quad \text{W/m}²

For peak-valued phasors, the time-average power density is

\langle\mathbf{S}\rangle=\frac{1}{2}\operatorname{Re}\{\mathbf{E}\times\mathbf{H}^*\}

For a lossless uniform plane wave with peak electric-field magnitude E₀,

\langle S\rangle=\frac{E₀²}{2\eta}

If RMS field values are used instead, the factor 1/2 is omitted. Mixing peak and RMS conventions is a common source of error.

2. Loss, Attenuation, and Skin Depth

A dielectric is often characterized by the loss tangent

\tan\delta=\frac{\sigma}{\omega\varepsilon}

A low-loss dielectric has \sigma\ll\omega\varepsilon. A good conductor has \sigma\gg\omega\varepsilon. For a good conductor,

\alpha\approx\beta\approx\sqrt{\frac{\omega\mu\sigma}{2}}=\sqrt{\pi f\mu\sigma}

The field magnitude decreases as e^{-\alpha z}. Skin depth is the distance over which the field falls to 1/e of its surface value:

\delta_s=\frac{1}{\alpha}\approx\sqrt{\frac{2}{\omega\mu\sigma}}=\frac{1}{\sqrt{\pi f\mu\sigma}}

This approximation applies only under the good-conductor condition. Skin depth decreases with increasing frequency, permeability, or conductivity. Consequently, high-frequency current crowds near conductor surfaces, increasing effective resistance.

Worked Example 1: Skin depth in copper

Find the skin depth of copper at 1 MHz. Take \sigma=5.8\times10⁷\ \text{S/m} and \mu\approx\mu₀=4\pi\times10^-7\ \text{H/m}.

\delta_s=\sqrt{\frac{2}{2\pi(10⁶)(4\pi\times10^-7)(5.8\times10⁷)}}

\delta_s\approx6.61\times10^-5\ \text{m}=66.1\ \mu\text{m}

Thus, most 1 MHz current flows within only a few tens of micrometres of the copper surface. Plating, surface roughness, and conductor geometry therefore matter in radio-frequency design.

3. Reflection and Refraction at Boundaries

At an interface, tangential electric and magnetic fields satisfy boundary conditions. In the absence of a surface current, tangential \mathbf{H} is continuous; in the absence of an imposed surface voltage or time-varying magnetic sheet, tangential \mathbf{E} is continuous.

At normal incidence from a lossless medium of impedance \eta₁ to another lossless medium of impedance \eta₂, the electric-field reflection and transmission coefficients are

\Gamma=\frac{E_r}{E_i}=\frac{\eta₂-\eta₁}{\eta₂+\eta₁}

\tau=\frac{E_t}{E_i}=\frac{2\eta₂}{\eta₁+\eta₂}=1+\Gamma

For real impedances, the reflected and transmitted power fractions are

R=|\Gamma|², \qquad T=\frac{\eta₁}{\eta₂}|\tau|²

For lossless media, R+T=1. A negative \Gamma indicates a 180° phase reversal of the reflected electric field.

Normal-incidence reflection and transmission at a planar boundary

Figure: Normal-incidence reflection and transmission at a planar boundary

Figure: At normal incidence, incident, reflected, and transmitted propagation directions lie on the same interface normal; the reflected wave returns in the opposite direction.

Worked Example 2: Air-to-dielectric interface

A wave in air strikes normally a lossless, nonmagnetic dielectric with \varepsilon_r=4. Its impedance is

\eta₂=\frac{\eta₀}{\sqrt{\varepsilon_r}}=\frac{376.73}{2}=188.37\ \Omega

Therefore,

\Gamma=\frac{188.37-376.73}{188.37+376.73}=-\frac{1}{3}

R=\frac{1}{9}=0.111, \qquad T=1-R=0.889

Approximately 11.1% of incident power is reflected and 88.9% transmitted. The reflected electric field undergoes phase reversal.

3.1 Oblique incidence

For oblique incidence, phase matching along the interface gives

\theta_r=\theta_i, \qquad \beta₁\sin\theta_i=\beta₂\sin\theta_t

For lossless nonmagnetic media this becomes Snell’s law, n₁\sin\theta_i=n₂\sin\theta_t. Reflection coefficients differ for transverse-electric (TE) and transverse-magnetic (TM) polarization. At the Brewster angle, TM reflection becomes zero for an interface between ordinary lossless dielectrics. Total internal reflection occurs when a wave travels from a higher phase index to a lower one and the incidence angle exceeds the critical angle.

4. Transmission Lines

At low frequencies, interconnections may be treated as lumped circuits. A distributed model is needed when line length is not negligible compared with wavelength or signal rise-time propagation distance. A common engineering rule is to consider transmission-line effects when physical length exceeds roughly one-tenth of the relevant wavelength, although fast digital edges may demand stricter treatment.

A uniform line has per-unit-length resistance R in Ω/m, inductance L in H/m, conductance G in S/m, and capacitance C in F/m. Its phasor-domain telegrapher equations are

\frac{dV}{dz}=-(R+j\omega L)I, \qquad \frac{dI}{dz}=-(G+j\omega C)V

The propagation constant and characteristic impedance are

\gamma=\sqrt{(R+j\omega L)(G+j\omega C)}

Z₀=\sqrt{\frac{R+j\omega L}{G+j\omega C}}

For an ideal lossless line, R=G=0, giving

Z₀=\sqrt{\frac{L}{C}}, \qquad v_p=\frac{1}{\sqrt{LC}}, \qquad \beta=\omega\sqrt{LC}

A load Z_L produces the load reflection coefficient

\Gamma_L=\frac{Z_L-Z₀}{Z_L+Z₀}

For a passive load on a lossless line, |\Gamma_L|\leq1. The voltage standing-wave ratio is

\text{VSWR}=\frac{1+|\Gamma_L|}{1-|\Gamma_L|}

A matched load has Z_L=Z₀, \Gamma_L=0, and VSWR = 1. Return loss is conventionally

\text{RL}=-20\log₁₀|\Gamma_L| \quad \text{dB}

Higher positive return loss indicates a better match. The input impedance of a line of length l is

Z_in=Z₀\frac{Z_L+Z₀\tanh(\gamma l)}{Z₀+Z_L\tanh(\gamma l)}

For a lossless line, this reduces to

Z_in=Z₀\frac{Z_L+jZ₀\tan(\beta l)}{Z₀+jZ_L\tan(\beta l)}

Worked Example 3: Mismatched 50 Ω line

A lossless 50 Ω line is terminated by 100 Ω.

\Gamma_L=\frac{100-50}{100+50}=\frac{1}{3}

\text{VSWR}=\frac{1+1/3}{1-1/3}=2

\text{RL}=-20\log₁₀(1/3)=9.54\ \text{dB}

If the line is one-quarter wavelength long, \beta l=\pi/2, and

Z_in=\frac{Z₀²}{Z_L}=\frac{50²}{100}=25\ \Omega

This impedance-transforming behavior is used in matching networks. The quarter-wave formula applies to a lossless line exactly one-quarter wavelength long, or an odd multiple thereof.

5. Rectangular Waveguides

A hollow metallic waveguide confines fields by reflection from conducting walls. It supports transverse-electric (TE) and transverse-magnetic (TM) modes but not a TEM mode because a hollow single-conductor guide cannot support the required two-conductor electrostatic potential structure.

For a rectangular waveguide of width a, height b, and homogeneous filling medium \mu,\varepsilon, the cutoff frequency of mode TE_mn or TM_mn is

f_c,mn=\frac{1}{2\sqrt{\mu\varepsilon}}\sqrt{\left(\frac{m}{a}\right)²+\left(\frac{n}{b}\right)²}

For TM modes, neither index may be zero. The dominant mode of a guide with a>b is TE₁₀, whose cutoff in air is f_c=c/(2a). Above cutoff,

\beta=\sqrt{\omega²\mu\varepsilon-k_c²}

\lambda_g=\frac{\lambda}{\sqrt{1-(f_c/f)²}}

The TE and TM wave impedances are

Z_TE=\frac{\omega\mu}{\beta}, \qquad Z_TM=\frac{\beta}{\omega\varepsilon}

These are modal wave impedances, not transmission-line characteristic impedances in the ordinary two-conductor sense.

Electric-field distribution of the rectangular-waveguide TE10 mode

Figure: Electric-field distribution of the rectangular-waveguide TE10 mode

Figure: In the TE₁₀ mode, the electric field is directed uniformly along the guide height, while its magnitude varies sinusoidally across the width.

Worked Example 4: WR-90 waveguide

For an air-filled WR-90 guide, a=22.86\ \text{mm}. The dominant-mode cutoff is

f_c=\frac{2.998\times10⁸}{2(0.02286)}=6.56\ \text{GHz}

At 10 GHz, the free-space wavelength is 29.98 mm. Therefore,

\lambda_g=\frac{29.98}{\sqrt{1-(6.56/10)²}}=39.7\ \text{mm}

The TE wave impedance is

Z_TE=\frac{\eta₀}{\sqrt{1-(f_c/f)²}}\approx499\ \Omega

Operation too close to cutoff produces large dispersion and a long guide wavelength. Practical systems commonly operate within a recommended single-mode band that avoids both cutoff and the next higher mode.

6. Antennas and Electromagnetic Radiation

An antenna converts guided electromagnetic energy into radiated waves and performs the reciprocal conversion during reception. Important antenna quantities include:

Quantity	Meaning
Radiation pattern	Angular distribution of radiated field or power
Directivity D	Concentration relative to an isotropic radiator
Gain G	Directivity multiplied by radiation efficiency
Polarization	Time-varying orientation of the electric field
Input impedance	Terminal voltage-to-current ratio
Bandwidth	Frequency range meeting specified performance
Effective aperture A_e	Receiving area equivalent for captured power

Gain and effective aperture are related by

A_e=\frac{G\lambda²}{4\pi}

For matched, polarization-aligned antennas in unobstructed free space, the Friis transmission equation is

P_r=P_tG_tG_r\left(\frac{\lambda}{4\pi R}\right)²

Here P_t and P_r are transmitted and received powers in watts, G_t and G_r are dimensionless power gains, and R is separation in metres. Friis applies in the far field and does not include obstruction, fading, feeder loss, impedance mismatch, or polarization mismatch unless correction factors are added.

Worked Example 5: Free-space radio link

A 2.4 GHz transmitter supplies 0.1 W to an antenna of 3 dBi gain. The receiving antenna also has 3 dBi gain and is 100 m away. Since 3 dBi corresponds to G=10^3/10=1.995, and \lambda=0.1249\ \text{m},

P_r=0.1(1.995)²\left(\frac{0.1249}{4\pi(100)}\right)²

P_r\approx3.93\times10^-9\ \text{W}\approx-54.1\ \text{dBm}

The result is an ideal free-space value. A real link budget must include cable losses, fading margin, receiver sensitivity, interference, and regulatory transmit-power limits.

6.1 Near and far fields

The reactive near field stores energy close to an antenna, whereas the radiating far field has locally transverse fields and approximately spherical spreading. For an antenna of maximum dimension D, a commonly used far-field criterion is

R\gtrsim\frac{2D²}{\lambda}

This criterion is most relevant when D is large relative to wavelength; additional distance conditions may be needed for electrically small antennas.

7. Electromagnetic Compatibility

EMC is the ability of equipment to operate satisfactorily in its electromagnetic environment without producing intolerable disturbance. An interference problem normally requires three elements:

a source of electromagnetic disturbance;
a coupling path;
a susceptible receiver or victim.

Coupling can be conducted through wires, capacitively coupled by electric fields, inductively coupled by magnetic fields, or radiated through space. Effective mitigation targets the source, path, or victim.

Mechanism	Typical mitigation
Conducted common-mode noise	Common-mode choke, feedthrough capacitor, bonding
Conducted differential-mode noise	Differential filter, reduced current-loop area
Capacitive coupling	Shielding, increased spacing, reduced dv/dt
Inductive coupling	Twisted pairs, smaller loops, reduced di/dt
Radiated coupling	Enclosure shielding, filtering, antenna relocation
Ground-related interference	Controlled return paths and appropriate bonding

Shielding effectiveness depends on reflection, absorption, seams, apertures, and cable penetrations. A conductive enclosure can perform poorly if an unfiltered cable carries common-mode current through it. EMC must therefore be designed at system level rather than added only after testing.

Good practice includes maintaining continuous return paths, separating noisy and sensitive circuits, minimizing loop areas, terminating high-speed traces correctly, filtering at enclosure boundaries, and verifying performance using calibrated pre-compliance measurements.

8. Spectrum, Safety, and Contemporary Applications

The radio spectrum is a shared and finite resource. National authorities coordinate frequency allocations under international frameworks developed through the International Telecommunication Union. Engineers must verify local rules governing frequency bands, occupied bandwidth, emission masks, channel access, equivalent isotropically radiated power, and out-of-band emissions.

Unlicensed operation does not mean unregulated operation. Devices may still be subject to power, duty-cycle, equipment-authorization, and interference-acceptance conditions. Safety assessment may also require compliance with limits on human exposure, expressed through quantities such as power density or specific absorption rate, depending on frequency and use.

Contemporary developments include massive multiple-input multiple-output systems, millimetre-wave communication, reconfigurable intelligent surfaces, ultra-wideband localization, software-defined radio, cognitive spectrum sensing, and satellite megaconstellations. Machine-learning methods can assist modulation recognition, anomaly detection, adaptive beamforming, and spectrum occupancy prediction. Such systems require representative data, robustness testing, cybersecurity controls, and safeguards against harmful autonomous transmissions.

9. Applied Case: Wireless Sensor Link in an Industrial Plant

This is an illustrative case.

Background

A manufacturing plant proposes a 915 MHz wireless vibration sensor mounted 120 m from a gateway. A variable-frequency motor drive lies between the units and produces broadband switching noise. The transmitter delivers 20 dBm to a 2 dBi antenna; the gateway antenna gain is 5 dBi. Cable loss totals 2 dB, and measured obstruction loss is estimated at 12 dB. Receiver sensitivity is −100 dBm.

Problem

The design must achieve reliable communication while preventing the motor drive and sensor electronics from causing regulatory or functional interference.

Analysis

The free-space path loss in decibels is

L_FS=20\log₁₀\left(\frac{4\pi R}{\lambda}\right)

At 915 MHz, \lambda=2.998\times10⁸/(915\times10⁶)=0.3277\ \text{m}. Thus,

L_FS=20\log₁₀\left(\frac{4\pi(120)}{0.3277}\right)=73.3\ \text{dB}

The predicted received power is

P_r=20+2+5-2-12-73.3=-60.3\ \text{dBm}

The nominal link margin is

M=-60.3-(-100)=39.7\ \text{dB}

The link has substantial nominal margin, but impulsive interference can still reduce packet reliability if receiver blocking or common-mode cable coupling occurs.

Approach and Solution

The engineering team:

relocates the gateway antenna away from the motor-drive cabinet;
uses shielded motor cable with circumferential shield termination;
installs input and output filtering at enclosure penetrations;
routes sensor and power wiring separately;
uses frequency-agile communication within the legally permitted channels;
performs conducted and radiated pre-compliance scans;
verifies permitted transmit power, channel access rules, and occupied bandwidth;
retains fading margin rather than increasing transmitter power unnecessarily.

Outcome

Testing shows that the received signal remains well above sensitivity and that packet errors caused by drive switching are greatly reduced. Pre-compliance measurements identify one cable resonance, corrected by improved bonding and a common-mode choke.

Learning Points

A large free-space link margin does not guarantee electromagnetic compatibility. Reliable design requires simultaneous consideration of propagation, antenna placement, interference coupling, receiver susceptibility, cable behavior, and spectrum regulation.

10. Engineering Challenges and Limitations

Electromagnetic models are approximations. Plane-wave analysis assumes fields sufficiently far from sources and interfaces large enough to neglect edge effects. Friis assumes far-field free-space propagation. Transmission-line models assume a sufficiently uniform cross-section, while simple waveguide formulas assume ideal geometry and homogeneous filling.

Real systems exhibit conductor roughness, dielectric dispersion, connector discontinuities, multipath fading, polarization mismatch, manufacturing tolerances, nonlinear devices, thermal drift, and uncertain grounding. Simulation is valuable but depends on geometry, mesh quality, material data, boundary conditions, and solver assumptions. Measurements should therefore validate important designs.

Professional practice also requires traceable calculations, calibrated instruments, documented test configurations, conservative margins, and compliance with applicable product, spectrum, exposure, and EMC standards.

Chapter Summary

Electromagnetic waves follow directly from Maxwell’s equations and are characterized by propagation constant, intrinsic impedance, wavelength, velocity, polarization, and power flow. Conducting media attenuate waves and exhibit skin effect, while impedance discontinuities cause reflection and transmission. Oblique incidence additionally involves refraction and polarization-dependent behavior.

Transmission lines are distributed systems described by the telegrapher equations, characteristic impedance, reflection coefficient, standing waves, and input impedance. Metallic waveguides propagate discrete TE and TM modes above cutoff; the dominant rectangular-guide mode is TE₁₀. Antennas convert guided energy to radiation, and free-space link behavior can be estimated using gain, effective aperture, and the Friis equation.

Successful engineering application requires more than ideal propagation calculations. EMC design must control sources, coupling paths, and susceptible circuits, while spectrum use must satisfy technical, safety, and regulatory constraints. Measurement, simulation, margins, and system-level design are complementary tools.

Key Terms

Attenuation constant: Exponential field-decay rate, measured in Np/m.

Characteristic impedance: Voltage-to-current ratio of a single traveling wave on a transmission line.

Cutoff frequency: Lowest frequency at which a specified waveguide mode propagates.

Electromagnetic compatibility: Ability of equipment to operate without causing or suffering unacceptable electromagnetic disturbance.

Far field: Region where radiation dominates and fields are approximately transverse.

Gain: Antenna directivity multiplied by radiation efficiency.

Intrinsic impedance: Electric-to-magnetic field ratio of a uniform plane wave in a medium.

Loss tangent: Ratio of conductive to displacement-current effects in a dielectric.

Mode: A permitted field configuration in a guiding structure.

Phase constant: Spatial rate of phase change, measured in rad/m.

Poynting vector: Electromagnetic power-flow density and direction.

Propagation constant: Complex quantity \gamma=\alpha+j\beta describing attenuation and phase progression.

Reflection coefficient: Complex reflected-to-incident wave-amplitude ratio.

Return loss: Logarithmic measure of mismatch based on reflection-coefficient magnitude.

Skin depth: Depth at which a field in a good conductor falls to 1/e of its surface value.

Standing wave: Spatial interference pattern produced by oppositely traveling waves.

TE mode: Guided mode with no electric-field component in the propagation direction.

Transmission line: Distributed structure that guides voltage and current waves.

VSWR: Ratio of maximum to minimum standing-wave voltage magnitude.

Waveguide: Structure that confines and directs electromagnetic waves through modal propagation.

Review Questions

Short Answer Questions

What physical role does displacement current play in electromagnetic-wave propagation?
Distinguish intrinsic impedance from transmission-line characteristic impedance.
Why does skin depth decrease as frequency increases?
Why can a hollow rectangular waveguide not support a TEM mode?
Identify the three elements required for an electromagnetic-interference problem.

Long Answer Questions

Derive the electric-field wave equation in a homogeneous conducting medium from Maxwell’s equations.
Explain normal-incidence reflection and transmission, including field and power coefficients.
Develop the telegrapher equations and explain the physical meanings of R, L, G, and C.
Describe rectangular-waveguide modes, cutoff, dispersion, guide wavelength, and dominant-mode operation.
Discuss antenna gain, effective aperture, polarization, far-field conditions, and limitations of the Friis equation.

Analytical / Application Questions

A 75 Ω lossless line is terminated by 25 Ω. Calculate \Gamma_L, VSWR, return loss, and the input impedance of a quarter-wave section.
An air-filled rectangular waveguide has a=30\ \text{mm} and b=15\ \text{mm}. Determine the first two cutoff frequencies and identify a suitable single-mode operating interval.
Develop an EMC improvement plan for a microcontroller board that resets whenever a nearby relay coil is switched. Address source suppression, coupling paths, grounding, filtering, layout, and verification.

Multiple Choice Questions with Answers

Question 1

A lossless dielectric has \varepsilon_r=9 and \mu_r=1. What is its intrinsic impedance?

A. 41.9 Ω

B. 125.6 Ω

C. 376.7 Ω

D. 1130 Ω

Correct Answer: B. 125.6 Ω

Question 2

For a good conductor, increasing frequency by a factor of four changes skin depth by what factor, assuming conductivity and permeability remain constant?

A. It doubles

B. It is unchanged

C. It becomes one-half

D. It becomes one-quarter

Correct Answer: C. It becomes one-half

Question 3

A wave traveling in a 50 Ω line reaches a 100 Ω load. What is the load reflection coefficient?

A. −1/3

B. 0

C. 1/3

D. 2

Correct Answer: C. 1/3

Question 4

The reflection-coefficient magnitude on a lossless line is 0.5. What is the VSWR?

A. 1.5

B. 2

C. 3

D. 4

Correct Answer: C. 3

Question 5

At normal incidence from a 376.7 Ω medium to a 188.4 Ω medium, what fraction of incident power is reflected?

A. 1/9

B. 1/3

C. 2/3

D. 8/9

Correct Answer: A. 1/9

Question 6

Which statement correctly describes the electric field of the rectangular-waveguide TE₁₀ mode?

A. It is longitudinal and constant across the width

B. It is directed across the height and varies sinusoidally across the width

C. It circulates around the guide axis with constant magnitude

D. It is zero everywhere along the guide height

Correct Answer: B. It is directed across the height and varies sinusoidally across the width

Question 7

An air-filled rectangular waveguide has broad dimension a=25\ \text{mm}. Approximately what is the TE₁₀ cutoff frequency?

A. 3.0 GHz

B. 6.0 GHz

C. 12.0 GHz

D. 25.0 GHz

Correct Answer: B. 6.0 GHz

Question 8

Under otherwise identical Friis-equation conditions, doubling antenna separation changes received power by what factor?

A. It becomes one-half

B. It becomes one-quarter

C. It doubles

D. It quadruples

Correct Answer: B. It becomes one-quarter

Question 9

Which action most directly reduces inductive coupling between a signal pair and a nearby current loop?

A. Increasing the signal-loop area

B. Removing the return conductor

C. Twisting the signal conductors together

D. Increasing relay current rise rate

Correct Answer: C. Twisting the signal conductors together

Question 10

Which condition is required for zero reflection at the load of a uniform transmission line?

A. Z_L=0

B. Z_L=Z₀

C. Z_L=2Z₀

D. Z_L=-Z₀

Correct Answer: B. Z_L=Z₀

Engineering Physics
Chapter 6 of 14

Chapter 6 Electromagnetic Waves, Transmission and Radiation