1. Electromagnetic Foundations of Antennas

Introduction and Context

An antenna is the interface between guided electromagnetic energy and electromagnetic waves that propagate through space. In a transmitting system, currents and voltages associated with a guided mode are transformed into fields that detach from the source and carry power to a remote location. In a receiving system, an incident electromagnetic field induces currents and voltages in a conducting structure, thereby coupling free-space energy into a transmission line or receiver. Antenna theory is therefore not an isolated topic; it connects Maxwell’s field equations, transmission-line theory, wave propagation, circuit concepts, materials, and communication systems.

The physical behavior of antennas is governed by the same laws that describe static electric and magnetic fields. The distinction is that time-varying fields can sustain one another and can transport energy over large distances. A complete understanding of antennas begins with the field equations, continues through material interfaces and wave propagation, and culminates in the description of radiation from time-varying sources.

This chapter develops that foundation. It explains Maxwell’s equations in differential and integral form, derives the wave equation, interprets boundary conditions, distinguishes near-field and far-field behavior, introduces polarization and power flow, and shows how retarded potentials describe radiation. Particular attention is given to the transition from guided waves to radiated fields, because this transition is central to the design of practical antennas.

Learning Objectives

After studying this chapter, the reader should be able to:

1. State and interpret Maxwell’s equations in differential and integral form.

2. Explain the constitutive relations and the roles of permittivity, permeability, and conductivity.

3. Apply electromagnetic boundary conditions at material interfaces.

4. Derive the wave equation and relate wave velocity, impedance, and wavelength to material properties.

5. Describe polarization, Poynting vector, power density, and electromagnetic energy flow.

6. Distinguish reactive near fields, radiating near fields, and the far field.

7. Explain how accelerated charges and time-varying currents produce radiation.

8. Interpret retarded potentials and their importance in antenna analysis.

9. Relate guided-wave modes to radiated fields at an antenna aperture or discontinuity.

10. Recognize practical, computational, and measurement issues associated with electromagnetic radiation.

Maxwell’s Equations: The Governing Structure

For macroscopic electromagnetic fields, Maxwell’s 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, \(\mathbf{E}\) is electric field intensity, \(\mathbf{H}\) is magnetic field intensity, \(\mathbf{D}\) is electric flux density, \(\mathbf{B}\) is magnetic flux density, \(\rho_v\) is volume charge density, and \(\mathbf{J}\) is conduction current density. The first equation is Gauss’s law for electricity: electric charge is the source of electric flux. The second is Gauss’s law for magnetism: magnetic flux has no experimentally observed isolated sources, so magnetic field lines form closed paths. Faraday’s law states that a time-varying magnetic field produces a circulating electric field. Ampère–Maxwell law states that magnetic circulation is produced both by conduction current and by time-varying electric flux.

The displacement-current term, \(\partial\mathbf{D}/\partial t\), is essential. Without it, charge conservation would be inconsistent with Ampère’s law in regions such as a capacitor gap, where conduction current is absent but electric flux is changing. Maxwell’s correction also permits electromagnetic waves to propagate through regions containing no conduction current.

The integral forms provide a complementary physical interpretation:

\[

\oint_S \mathbf{D}\cdot d\mathbf{S}=Q_{enc}, \qquad

\oint_S \mathbf{B}\cdot d\mathbf{S}=0,

\]

\[

\oint_C \mathbf{E}\cdot d\mathbf{l}=-\frac{d}{dt}\int_S\mathbf{B}\cdot d\mathbf{S},

\]

\[

\oint_C \mathbf{H}\cdot d\mathbf{l}=I_{\rm enc}+\frac{d}{dt}\int_S\mathbf{D}\cdot d\mathbf{S}.

\]

The differential form is particularly useful for local field analysis, whereas the integral form is valuable for symmetry arguments, circuit interpretations, and boundary-condition derivations.

Constitutive Relations and Material Parameters

In a linear, isotropic, homogeneous medium,

\[

\mathbf{D}=\epsilon\mathbf{E},\qquad \mathbf{B}=\mu\mathbf{H},\qquad \mathbf{J}=\sigma\mathbf{E},

\]

where \(\epsilon\) is permittivity, \(\mu\) is permeability, and \(\sigma\) is conductivity. In anisotropic materials these quantities are tensors rather than scalars. In dispersive materials they depend on frequency, and in lossy media they may be represented through complex parameters.

The total current density may be written as

\[

\mathbf{J}_{\rm total}=\sigma\mathbf{E}+\frac{\partial\mathbf{D}}{\partial t}.

\]

The ratio \(\sigma/(\omega\epsilon)\) indicates the relative importance of conduction and displacement currents at angular frequency \(\omega\). A good conductor has \(\sigma\gg\omega\epsilon\), while a low-loss dielectric has \(\sigma\ll\omega\epsilon\). This distinction affects attenuation, penetration depth, antenna efficiency, and the distribution of fields near conductors.

Boundary Conditions at Interfaces

Antenna structures frequently contain interfaces between conductors, dielectrics, magnetic materials, and free space. The fields cannot be analyzed correctly without applying interface conditions.

At an interface with unit normal \(\hat{\mathbf{n}}\) directed from medium 1 to medium 2, the general boundary conditions are

\[

\hat{\mathbf{n}}\cdot(\mathbf{D}_2-\mathbf{D}_1)=\rho_s,

\]

\[

\hat{\mathbf{n}}\cdot(\mathbf{B}_2-\mathbf{B}_1)=0,

\]

\[

\hat{\mathbf{n}}\times(\mathbf{E}_2-\mathbf{E}_1)=\mathbf{0},

\]

\[

\hat{\mathbf{n}}\times(\mathbf{H}_2-\mathbf{H}_1)=\mathbf{J}_s,

\]

where \(\rho_s\) is surface charge density and \(\mathbf{J}_s\) is surface current density. Thus, the normal component of electric flux changes according to surface charge, the normal magnetic flux is continuous, the tangential electric field is continuous in the absence of an impressed singular electromotive force, and the tangential magnetic field changes according to surface current.

For a perfect electric conductor (PEC), the tangential electric field at the surface is zero:

\[

\mathbf{E}_t=0.

\]

The magnetic field may have a tangential component related to surface current by \(\mathbf{J}_s=\hat{\mathbf{n}}\times\mathbf{H}\). The electric field may have a normal component associated with surface charge. These conditions explain why currents flow along the surfaces of conductors and why an antenna’s geometry strongly determines its current distribution.

At a dielectric interface without free surface charge or surface current, tangential components of both fields are continuous. A wave incident on such an interface generally produces reflected and transmitted fields. In an antenna, this phenomenon appears in dielectric radomes, substrates, superstrates, and layered media.

Wave Propagation in a Uniform Medium

Consider a source-free, homogeneous, isotropic medium. With \(\mathbf{J}=0\) and \(\rho_v=0\), Maxwell’s equations lead to

\[

\nabla^2\mathbf{E}-\mu\epsilon\frac{\partial^2\mathbf{E}}{\partial t^2}=0,

\]

for a lossless medium. Similarly,

\[

\nabla^2\mathbf{H}-\mu\epsilon\frac{\partial^2\mathbf{H}}{\partial t^2}=0.

\]

The phase velocity is

\[

v=\frac{1}{\sqrt{\mu\epsilon}},

\]

and the intrinsic impedance is

\[

\eta=\sqrt{\frac{\mu}{\epsilon}}.

\]

In free space, \(v=c\approx3\times10^8\) m/s and \(\eta_0\approx377\ \Omega\). For a uniform plane wave traveling in the \(+z\) direction, if \(\mathbf{E}=\hat{\mathbf{x}}E_0\cos(\omega t-\beta z)\), then \(\mathbf{H}=\hat{\mathbf{y}}(E_0/\eta)\cos(\omega t-\beta z)\). The three vectors \(\mathbf{E}\), \(\mathbf{H}\), and the direction of propagation are mutually perpendicular.

The phase constant is \(\beta=\omega\sqrt{\mu\epsilon}\), and wavelength is \(\lambda=2\pi/\beta=v/f\). In a conducting medium, the wave number is complex. The field both propagates and attenuates, and the attenuation constant determines penetration into the material. This is important in shielding, ground-wave propagation, biomedical antennas, and antenna platforms containing conductive structures.

A plane wave is an idealization. Practical antennas generate spherical or approximately spherical wavefronts near the source. Nevertheless, the plane-wave model becomes increasingly accurate in the far field, where the wavefront curvature over the observation region is small.

Polarization and the Geometry of Fields

Polarization describes the time-varying orientation of the electric-field vector at a fixed observation point. If the field has only one component, it is linearly polarized. If two orthogonal components have equal amplitudes and a phase difference of \(90^\circ\), the field is circularly polarized. More general amplitude and phase relationships produce elliptical polarization.

For a field

\[

\mathbf{E}(t)=\hat{\mathbf{x}}E_x\cos\omega t+\hat{\mathbf{y}}E_y\cos(\omega t+\delta),

\]

the polarization depends on \(E_x\), \(E_y\), and \(\delta\). Polarization matching affects received power. A linearly polarized receiving antenna aligned with an incident linear field can receive maximum power; a ninety-degree misalignment ideally produces zero coupling. Circularly polarized antennas are useful where orientation changes or multipath can cause polarization rotation, such as satellite links and navigation systems.

Polarization is a field property, not merely a mechanical description of antenna shape. A physically tilted antenna does not automatically have arbitrary polarization, and near-field polarization may differ significantly from far-field polarization. For this reason, polarization should be evaluated at the observation region relevant to the application.

Power Flow, Energy Storage, and Radiation

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

\[

\mathbf{S}=\mathbf{E}\times\mathbf{H}.

\]

For sinusoidal fields represented by phasors, the time-average power density is

\[

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

\]

The direction of the real part indicates net outward or inward power transport. The imaginary part of complex power is associated with reactive energy that is stored and returned during each cycle. Near an antenna, electric and magnetic energy may oscillate locally without contributing to net radiation. Far from the antenna, the fields become approximately transverse, \(|\mathbf{E}|/|\mathbf{H}|\to\eta_0\), and the power flow is predominantly outward.

The complex Poynting theorem expresses energy conservation. In phasor form, it relates the net electromagnetic power leaving a volume to source power, ohmic loss, and reactive energy storage. This theorem provides the basis for distinguishing accepted power, radiated power, conductor and dielectric loss, and reflected power. Antenna efficiency is fundamentally an energy-accounting quantity:

\[

\eta_{\rm rad}=\frac{P_{\rm rad}}{P_{\rm accepted}}.

\]

Antenna gain further includes the effect of directivity, while directivity describes the concentration of radiation relative to an isotropic radiator. These quantities are developed in detail in later antenna chapters, but their physical meaning begins with the Poynting vector.

Why Time-Varying Currents Radiate

A static charge produces a static electric field, and a steady current produces a static magnetic field. Radiation requires time variation. More specifically, radiation is associated with field components that detach from the source and carry energy to infinity with amplitudes decreasing approximately as \(1/r\). Terms decreasing as \(1/r^2\) or \(1/r^3\) are near-field terms: they may store energy and diminish more rapidly with distance.

An accelerating charge radiates because its electromagnetic field cannot adjust instantaneously throughout space. A time-varying current in a conductor produces changing electric and magnetic fields. If the current distribution is arranged so that the fields do not cancel completely at distant points, a radiated wave results. This is why a straight wire carrying a uniform direct current does not radiate appreciably, whereas an alternating current in a finite dipole does.

The phrase “current element radiates” is a useful mathematical model, but a physically realizable antenna must satisfy charge continuity and boundary conditions. Radiation is determined by the complete current and charge distribution, including return currents and feed structures. Neglecting these features can lead to incorrect predictions of pattern, polarization, or impedance.

Retarded Potentials and the Origin of Radiation Fields

Vector and scalar potentials provide a convenient route to solving radiation problems. The magnetic vector potential \(\mathbf{A}\) and electric scalar potential \(\phi\) are defined by

\[

\mathbf{B}=\nabla\times\mathbf{A},

\qquad

\mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}.

\]

For a localized time-varying source in free space, the potentials depend on the source at an earlier time,

\[

t_r=t-\frac{R}{c},

\]

called the retarded time. In schematic form,

\[

\mathbf{A}(\mathbf{r},t)=\frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}',t-R/c)}{R}\,dv',

\]

where \(R=|\mathbf{r}-\mathbf{r}'|\). The scalar potential similarly depends on retarded charge density. The delay is not a mathematical inconvenience; it represents the finite propagation speed of electromagnetic influence.

For sinusoidal sources, the retarded dependence becomes a phase factor \(e^{-jkR}\). In the far zone, where the observation distance is much larger than the source dimensions, approximations simplify the integral. The amplitude factor becomes approximately \(1/r\), and the phase variation across the source determines the radiation pattern. This is the origin of the array factor and aperture-field concepts used throughout antenna engineering.

The fields derived from the potentials contain terms proportional to \(1/r^3\), \(1/r^2\), and \(1/r\). The first two are associated primarily with electrostatic and induction effects; the \(1/r\) terms are radiation fields. In the far field, the radiation electric and magnetic fields are transverse and related by the intrinsic impedance of the surrounding medium.

From Guided Waves to Radiated Fields

In a transmission line or waveguide, electromagnetic energy is confined by conductors or dielectric boundaries. The fields form a guided mode with a specific propagation constant, field distribution, impedance, and power flow. A guided wave does not normally radiate because its fields satisfy boundary conditions that confine energy to the guiding structure.

Radiation begins when the guided mode encounters a discontinuity or an open structure that permits fields to extend into an unbounded region. Examples include:

• the open end of a waveguide;
• the transition from a coaxial line to a monopole;
• a microstrip feed connected to a patch antenna;
• a slot or aperture in a conducting surface;
• a rapidly changing bend, termination, or impedance discontinuity.

At the transition, the tangential fields and currents must satisfy Maxwell’s boundary conditions. The guided field is redistributed into bound, evanescent, and radiating components. A portion may be reflected, a portion dissipated, and a portion converted into outward-propagating waves. Efficient antenna design seeks a controlled transition in which the desired radiating mode is excited while unwanted reflection and loss are minimized.

An aperture antenna illustrates this process clearly. A waveguide mode reaches an aperture and produces electric and magnetic fields over the opening. Those fields act as an equivalent source for the external radiation problem. The far-field pattern is determined largely by the amplitude and phase distribution over the aperture. In a microstrip antenna, currents on the patch and ground plane create fringing fields at the edges; these fringing fields constitute the principal radiating mechanism.

The transition is not merely an impedance event. It is also a spatial transformation: a confined field distribution becomes a radiating field distribution. Matching, mode conversion, aperture shape, current continuity, and phase distribution must therefore be considered together.

Regions Around an Antenna

The field surrounding a finite antenna is commonly divided into regions, although exact boundaries depend on antenna size and shape. The reactive near-field region lies closest to the antenna and is dominated by stored electric and magnetic energy. Its fields can vary rapidly with position and may contain strong non-radiating components.

The radiating near-field, or Fresnel region, is farther away. Radiation is present, but angular field distribution may still depend on distance because phase differences across the antenna aperture are significant. The far-field, or Fraunhofer region, is sufficiently distant that the angular pattern is essentially independent of range. A widely used engineering criterion for the beginning of the far field is

\[

r\gtrsim \frac{2D^2}{\lambda},

\]

where \(D\) is the largest antenna dimension. This criterion should be applied with judgment, particularly for arrays, electrically large reflectors, near-field measurement systems, and broadband antennas.

In the far field, \(\mathbf{E}\) and \(\mathbf{H}\) are transverse, their ratio approaches the medium impedance, and power density decreases approximately as \(1/r^2\). The total power through a sphere remains approximately constant because the sphere’s area grows as \(r^2\).

Applied Case: Coaxial Feed to a Monopole

Background

Consider a quarter-wave monopole mounted on a conducting ground plane and fed by a coaxial cable. The coaxial cable supports a transverse electromagnetic mode in which electric and magnetic fields are confined between the inner and outer conductors.

Situation and Problem

At the connector, the inner conductor continues upward as the monopole while the outer conductor is bonded to the ground plane. The design problem is to understand how the guided coaxial mode becomes a radiated field and why unwanted common-mode current on the outside of the coax can distort the radiation pattern.

Analysis

Within the coax, the equal and opposite currents on the inner and inner surface of the outer conductor produce a confined field. At the monopole termination, the current cannot continue in the same guided configuration. Charge accumulates near the open end, and the current distribution varies along the monopole. The conducting ground plane provides the return-current path and acts, through image theory under suitable conditions, as the counterpart of a dipole arm.

The vertical current produces an electric field with predominantly vertical polarization in the far zone. The time-varying current and charge distribution generate radiation fields, while nearby reactive fields store energy. If current flows on the outer surface of the coaxial shield, the feed cable itself becomes part of the radiator. The resulting pattern may become asymmetric, and the measured impedance may differ from the simulated value obtained using an idealized feed.

Approach and Solution

A practical design uses a sufficiently large or well-characterized ground plane, a controlled feed transition, and a common-mode suppression arrangement such as a choke or sleeve balun where appropriate. Simulation should include the actual feed region, connector, ground plane, and nearby dielectric supports. Measurement should verify both input impedance and three-dimensional pattern.

Outcome

When the transition is properly designed, most accepted power is delivered to the monopole and ground-plane current system, producing the intended omnidirectional azimuthal pattern. Suppressing feed current improves pattern symmetry and reduces uncertainty in gain and polarization.

Learning Points

The case demonstrates that radiation is produced by the complete current distribution, not by an isolated visible portion of the antenna. A feed is an electromagnetic component, and the guided-to-radiated transition must be analyzed using fields, currents, boundary conditions, and power flow together.

Practical and Professional Considerations

Several common mistakes recur in antenna analysis. Treating an antenna as a lumped circuit alone hides spatial phase and current-distribution effects. Assuming that every field near an antenna is radiating leads to incorrect estimates of power and safety margins. Using far-field formulas in the reactive region produces unreliable results. Omitting the feed cable, mounting structure, radome, or ground plane from a model can substantially change the result.

Numerical electromagnetic tools such as the method of moments, finite-element method, finite-difference time-domain method, and physical-optics approximations are valuable, but they do not replace physical reasoning. Mesh resolution, absorbing boundaries, port definitions, material models, and convergence testing must be selected consistently with frequency and geometry. Results should be checked against conservation of power, symmetry, limiting cases, and measurements.

Professional practice also requires attention to exposure limits, electromagnetic compatibility, spectrum regulations, and safe operation near high-power radiators. Antenna systems may interfere with sensitive equipment, implantable devices, navigation systems, or communication networks. Design decisions should therefore consider not only efficiency and pattern but also compliance, coexistence, environmental conditions, and maintainability.

Current Developments

Modern antenna analysis increasingly combines full-wave simulation, optimization, automated parameter sweeps, and measurement data. Machine-learning methods can assist surrogate modeling and design-space exploration, but they remain dependent on the quality and physical coverage of training data. Reconfigurable antennas use switches, varactors, microelectromechanical devices, or tunable materials to alter frequency, polarization, or pattern. Metasurfaces and engineered impedance surfaces provide additional control over phase and wave transformation. Despite these developments, all such systems remain subject to Maxwell’s equations, passivity and loss constraints, boundary conditions, and energy conservation.

Near-field scanning and computational near-field-to-far-field transformations are increasingly important for electrically large antennas and satellite payloads. These methods illustrate a continuing engineering principle: fields may be measured or computed in one region and transformed into another only when the underlying electromagnetic model and measurement conditions are properly controlled.

Best Practices for Foundation-Level Antenna Analysis

1. Begin with Maxwell’s equations and identify the relevant sources, materials, and boundaries.

2. Separate guided, reactive, evanescent, and radiating field components conceptually.

3. Use phasor notation only after defining the time convention.

4. Track accepted, reflected, dissipated, stored, and radiated power.

5. Include the complete electromagnetic structure, especially feeds and return-current paths.

6. Check polarization and coordinate conventions before interpreting patterns.

7. Confirm that the observation distance satisfies the assumptions of the chosen field approximation.

8. Validate analytical or numerical results using limiting cases, power balance, and measurement.

Chapter Summary

Maxwell’s equations provide the unified foundation for antenna theory. Constitutive relations connect fields to material properties, while boundary conditions determine how fields and currents behave at conductors and interfaces. In a source-free medium, Maxwell’s equations produce electromagnetic waves whose velocity, impedance, and wavelength depend on permittivity and permeability.

Time-varying currents and accelerating charges generate fields that include reactive, induction, and radiation terms. The radiation terms decrease as \(1/r\), carry power to distant locations, and become transverse in the far field. Polarization describes the trajectory of the electric-field vector, while the Poynting vector describes electromagnetic power flow. Retarded potentials account for the finite propagation time from source to observation point and provide a rigorous basis for radiation integrals.

An antenna operates by transforming a guided mode into a radiating field or, in reception, by performing the reverse transformation. This transition depends on boundary conditions, current distribution, aperture fields, matching, and feed structure. Sound antenna engineering therefore requires simultaneous attention to fields, sources, geometry, power, materials, computation, measurement, and regulatory responsibility.

Key Terms

Maxwell’s equations: The four fundamental equations governing macroscopic electric and magnetic fields.

Permittivity: A material parameter relating electric flux density to electric field intensity.

Permeability: A material parameter relating magnetic flux density to magnetic field intensity.

Conductivity: A material property describing conduction current produced by an electric field.

Displacement current: The term \(\partial\mathbf{D}/\partial t\) that accounts for time-varying electric flux in Ampère–Maxwell law.

Boundary condition: A field relationship that must hold at the interface between two electromagnetic media.

Perfect electric conductor: An ideal conductor whose tangential electric field at its surface is zero.

Intrinsic impedance: The ratio of electric-field magnitude to magnetic-field magnitude for a uniform plane wave.

Phase constant: The rate of phase change with distance, commonly denoted by \(\beta\).

Polarization: The time-dependent orientation and trajectory of the electric-field vector.

Poynting vector: The vector \(\mathbf{E}\times\mathbf{H}\), representing instantaneous electromagnetic power-flow density.

Reactive field: A field component associated primarily with local energy storage and return rather than net radiation.

Radiation field: The component that transports electromagnetic power outward and varies approximately as \(1/r\).

Retarded time: The source time delayed by the finite propagation interval from source to observation point.

Retarded potential: A scalar or vector potential evaluated using source values at retarded time.

Guided wave: An electromagnetic mode confined and directed by conductors or dielectric boundaries.

Aperture field: The electromagnetic field distribution over an opening that contributes to radiation.

Far field: The region in which the radiation pattern is essentially independent of distance and fields are approximately transverse.

Radiation efficiency: The ratio of radiated power to accepted antenna power.

Common-mode current: Current flowing in the same effective direction on conductors that ideally should carry balanced return currents, often causing unintended radiation.

Review Questions

Short Answer Questions

1. State Maxwell’s four equations in differential form and identify the source term in each relevant equation.

2. What is the physical significance of displacement current?

3. Define intrinsic impedance and phase velocity for a lossless uniform medium.

4. What is meant by retarded time?

5. Distinguish reactive near-field energy from radiated power.

Long Answer Questions

1. Derive the electromagnetic wave equation in a source-free, homogeneous, lossless medium and interpret each resulting parameter.

2. Explain the electromagnetic boundary conditions at a conductor–dielectric interface and apply them to a perfect electric conductor.

3. Discuss polarization types and explain how polarization mismatch affects a receiving antenna.

4. Explain how retarded potentials lead to \(1/r\) radiation fields and distinguish these from \(1/r^2\) and \(1/r^3\) terms.

5. Describe in detail the transition from a guided wave in a waveguide or coaxial line to a radiated field at an antenna structure.

Analytical / Application Questions

1. A plane wave travels through a lossless dielectric with relative permittivity 4 and relative permeability 1. Determine its phase velocity and intrinsic impedance relative to free space.

2. A reflector antenna has maximum dimension 2 m and operates at 10 GHz. Estimate the minimum distance for a conventional far-field measurement using \(r\approx2D^2/\lambda\), and explain the assumptions behind the estimate.

3. A monopole has the expected input impedance in simulation but exhibits an asymmetric measured pattern. Using the concepts in this chapter, identify likely causes and propose a measurement or design strategy to diagnose them.

22. Multiple Choice Questions with Answers

Question 1

Which Maxwell equation directly expresses the absence of isolated magnetic charges?

A. Gauss’s law for electricity

B. Gauss’s law for magnetism

C. Faraday’s law

D. Ampère–Maxwell law

Correct Answer: B. Gauss’s law for magnetism

Question 2

At the surface of an ideal perfect electric conductor, which condition must hold?

A. Normal magnetic field is zero in every adjacent medium

B. Tangential electric field is zero

C. Tangential magnetic field is zero

D. Normal electric flux density is always zero

Correct Answer: B. Tangential electric field is zero

Question 3

A field component proportional to 1/r in the distant region of an antenna is primarily associated with

A. Static electric storage

B. Static magnetic storage

C. Radiation carrying power to infinity

D. Conduction loss only

Correct Answer: C. Radiation carrying power to infinity

Question 4

For a uniform plane wave in a lossless medium, the ratio of electric-field magnitude to magnetic-field magnitude is the medium’s

A. Propagation constant

B. Intrinsic impedance

C. Wavelength

D. Conductivity

Correct Answer: B. Intrinsic impedance

Question 5

What does retarded time represent in an antenna radiation integral?

A. The time at which the receiver switches on

B. The source time delayed by propagation from source to observation point

C. The time required for a conductor to become charged

D. The time averaged over one period

Correct Answer: B. The source time delayed by propagation from source to observation point

Question 6

Which statement best describes the far field of an antenna?

A. Reactive energy dominates and fields vary mainly as 1/r^3

B. Electric and magnetic fields are approximately transverse and power density varies as 1/r^2

C. The electric field is always zero

D. The field is independent of frequency

Correct Answer: B. Electric and magnetic fields are approximately transverse and power density varies as 1/r^2

Question 7

A coaxial cable begins radiating unintentionally when significant current flows on its outer surface. This is an example of

A. Displacement-current cancellation

B. Common-mode current radiation

C. Perfect magnetic shielding

D. Polarization extinction

Correct Answer: B. Common-mode current radiation

Question 8

Two orthogonal electric-field components of equal amplitude with a 90-degree phase difference produce

A. Linear polarization

B. Circular polarization

C. Zero polarization

D. Purely reactive polarization

Correct Answer: B. Circular polarization

Question 9

Why does an open-ended waveguide radiate?

A. The guided mode encounters an opening that permits fields to couple into unbounded space

B. The waveguide loses all magnetic flux at the metal wall

C. The frequency becomes zero at the aperture

D. The displacement current disappears inside the waveguide

Correct Answer: A. The guided mode encounters an opening that permits fields to couple into unbounded space

Question 10

For a finite antenna, the commonly used estimate r ≥ 2D²/λ is primarily used to identify the beginning of the

A. Electrostatic region

B. Reactive near-field region

C. Conventional far-field region

D. Conductor interior

Correct Answer: C. Conventional far-field region