Electromagnetic Waves and Propagation

1

Electric field intensity (E) is the force per unit positive charge at a point: E = F/q, a vector quantity measured in V/m.

2

Electric flux density (D) relates to E via D = εE; Gauss's law states ∮D·dS = Q_enclosed.

3

Divergence of a vector field at a point measures the net outward flux per unit volume — positive divergence indicates a source, negative indicates a sink.

4

Divergence theorem (Gauss's theorem): ∮A·dS = ∫(∇·A)dv — converts a surface integral into a volume integral of the divergence, linking Gauss's law's integral and differential forms.

5

Electric potential (V) is the work done per unit charge to bring a test charge from infinity to a point, against the field; E = −∇V (potential gradient).

6

Energy density stored in an electrostatic field is w_E = ½εE², the energy stored per unit volume.

7

An electric dipole consists of two equal and opposite charges separated by a small distance, characterized by dipole moment p = Qd; its field falls off faster than a single point charge's field.

8

Polarization (P) is the dipole moment per unit volume induced in a dielectric by an applied field; it relates D and E via D = ε₀E + P.

9

**Free charges** can move freely (as in conductors), while **bound charges** are locked in place within the atoms/molecules of a dielectric but can shift slightly (polarize) under an applied field.

10

Relative permittivity (ε_r) is the ratio of a material's permittivity to that of free space, indicating how strongly it polarizes under an applied field.

11

The continuity equation, ∇·J = −∂ρ/∂t, expresses conservation of charge by relating current density divergence to the rate of decrease of charge density.

12

Relaxation time (τ = ε/σ) is the time constant for excess charge introduced into a conductor to dissipate to its surface.

13

Laplace's equation (∇²V = 0) governs electrostatic potential in a charge-free region, while Poisson's equation (∇²V = −ρ/ε) applies to a region with charge.

14

The uniqueness theorem guarantees that a solution satisfying the given boundary conditions is the only correct solution to a boundary value problem.

1

Biot-Savart's law gives the magnetic field dH produced by a small current element: dH = (I dl × a_R)/(4πR²) — the magnetic analog of Coulomb's law, useful for arbitrary current distributions.

2

Ampere's circuital law: ∮H·dl = I_enclosed — the line integral of H around a closed path equals the enclosed current; simplifies field calculation for symmetric current distributions.

3

Curl of a vector field at a point measures the field's tendency to rotate (circulate) around that point — a non-zero curl indicates a rotational (non-conservative) field.

4

Stoke's theorem: ∮A·dl = ∫(∇×A)·dS — converts a closed line integral into a surface integral of the curl, connecting Ampere's law's integral and differential forms.

5

The magnetic force on a moving charge is F = qv × B (the magnetic component of the Lorentz force); this force is perpendicular to both velocity and field, so it does no work on the charge.

6

The magnetic force on a current-carrying conductor is F = IL × B.

7

Magnetic torque: T = m × B, where m is the magnetic dipole moment; the torque tends to align the dipole with the field.

8

Magnetic dipole moment (m): m = IA, for a current loop of area A carrying current I.

9

Magnetization (M) is the magnetic dipole moment per unit volume induced in a material by an applied field; it relates B and H via B = μ₀(H+M).

10

At the interface between two different media, the tangential component of H is continuous (unless a surface current exists).

11

The normal component of B is always continuous across any boundary between two media, regardless of surface currents.

12

This section's magnetostatics fundamentals parallel electrostatics: Biot-Savart's and Ampere's laws mirror Coulomb's law and Gauss's law, while curl and Stoke's theorem mirror divergence and Gauss's (divergence) theorem.

1

Displacement current (J_d = ∂D/∂t) was introduced by Maxwell to fix an inconsistency in Ampere's law for time-varying fields (e.g., a capacitor gap where no conduction current flows), ensuring current continuity even without physical charge flow.

2

Gauss's law (electric), point form: ∇·D = ρ — electric flux diverges from free charge.

3

Gauss's law (magnetic), point form: ∇·B = 0 — no isolated magnetic monopoles exist.

4

Faraday's law, point form: ∇×E = −∂B/∂t — a time-varying magnetic field induces a circulating electric field.

5

Ampere's law (with Maxwell's correction): ∇×H = J + ∂D/∂t — the magnetic field circulates around conduction current plus displacement current.

6

In free space / lossless dielectric, a wave propagates without attenuation; the propagation constant is real-valued and E and H remain in phase.

7

In a lossy dielectric, a wave attenuates as it propagates due to finite conductivity; the propagation constant is complex, γ = α + jβ (α = attenuation constant, β = phase constant).

8

In a good conductor, a wave attenuates very rapidly (skin effect); E and H are 45° out of phase, and the wave is confined to a thin surface layer (skin depth δ).

9

**Normal incidence** is a plane wave striking a boundary perpendicularly; reflection/transmission coefficients depend on the intrinsic impedances of the two media.

10

**Oblique incidence** is a wave striking a boundary at an angle, governed by Snell's law, with distinct behavior for parallel and perpendicular polarization.

11

The Brewster angle is the specific angle of oblique incidence at which reflection is zero for parallel polarization.

12

Total internal reflection occurs beyond the critical angle, when going from a denser to a less dense medium.

1

A rectangular waveguide is a hollow conducting pipe (typically rectangular cross-section) that guides electromagnetic waves, used mainly at microwave frequencies where coaxial cable losses become excessive.

2

Unlike coaxial cable/twin-wire lines, a waveguide cannot support a pure TEM (Transverse Electromagnetic) mode; it supports only TE and TM modes.

3

In TE (Transverse Electric) mode, the electric field has no component in the direction of propagation (E_z=0); the magnetic field has a component along the propagation direction.

4

In TM (Transverse Magnetic) mode, the magnetic field has no component in the direction of propagation (H_z=0); the electric field has a component along the propagation direction.

5

Each mode has a cutoff frequency below which it cannot propagate in the guide.

6

The dominant mode for a rectangular waveguide is TE₁₀, which has the lowest cutoff frequency of all modes.

7

An antenna is a transducer converting guided electromagnetic energy (from a transmission line) into free-space radiation, and vice versa for reception.

8

The reciprocity theorem states that an antenna's transmitting and receiving properties (radiation pattern, gain, impedance) are identical.

9

Radiation pattern: a graphical representation of an antenna's radiated power (or field) as a function of direction in space.

10

Gain: the ratio of the maximum radiation intensity from the antenna to that of a reference (usually isotropic) antenna radiating the same total power.

11

Directivity is similar to gain but based on radiated power alone (excludes antenna losses); gain = efficiency × directivity.

12

Beamwidth: the angular width of the main lobe of the radiation pattern, typically measured at the half-power (3 dB) points.

13

Antenna impedance is the impedance presented by the antenna terminals, ideally matched to the feed line for maximum power transfer.

14

Polarization is the orientation of the electric field vector of the radiated wave (linear, circular, or elliptical); bandwidth is the range of frequencies over which the antenna's performance meets specified requirements.

1

An isotropic antenna is a hypothetical, idealized antenna radiating equally in all directions — a theoretical reference only, not physically realizable.

2

An omnidirectional antenna (e.g., a simple dipole) radiates uniformly in all directions within one plane (e.g., azimuth), but not uniformly in the perpendicular plane.

3

A directional antenna radiates (or receives) more effectively in specific directions, concentrating energy into a narrower beam for higher gain in that direction.

4

Traveling wave antennas (single wire, V antenna, Rhombus antenna) are non-resonant, terminated structures along which a wave travels in essentially one direction, producing a directional beam useful over a wide bandwidth; commonly used for HF communication.

5

A large/small plane sheet reflector is a flat conducting surface placed behind a driven element to redirect radiation forward, improving directivity.

6

A corner reflector uses two flat sheets meeting at an angle (often 90°), focusing radiation from a driven element into a narrower beam.

7

A parabolic reflector is a curved dish that focuses incoming parallel rays to a single feed point (or vice versa, from a feed to a parallel beam); it offers very high gain, common for satellite/microwave links.

8

Elliptical/hyperbolic/circular reflectors are specialized curved reflector geometries used in dual-reflector and specialized microwave antenna systems.

9

An aperture antenna (horn) is a flared waveguide section that gradually transitions the waveguide's impedance to that of free space, providing moderate gain with a simple, broadband design.

10

Array antennas combine multiple individual radiating elements so their fields interfere constructively in desired directions, increasing overall gain/directivity.

11

The Yagi-Uda antenna uses a driven element plus parasitic reflector and director elements, giving high gain and simple construction (common for TV reception).

12

The log-periodic antenna is a self-similar, frequency-independent array design offering wide bandwidth with consistent performance.

13

A monopole is a single vertical element over a ground plane (effectively half of a dipole); a loop antenna is a closed conducting loop, useful for magnetic-field sensing/direction finding.

14

A helical antenna is a helix-shaped conductor radiating circular polarization, used for satellite communication; a microstrip (patch) antenna is a flat conductive patch on a dielectric substrate, low-profile and easily integrated onto PCBs.

1

Ground/surface wave propagation: the wave travels along the Earth's surface, following its curvature to some extent.

2

Ground/surface wave is dominant at low/medium frequencies (e.g., AM broadcast), and attenuates with distance and ground conductivity.

3

Space wave propagation is a combination of a direct (line-of-sight) wave and a ground-reflected wave.

4

Space wave is the dominant mode for VHF/UHF and above, limited essentially to line-of-sight range.

5

Ionospheric (sky) wave: the wave is radiated upward and refracted (bent) back to Earth by the ionosphere's ionized layers.

6

Sky wave propagation enables long-distance HF communication well beyond the horizon.

7

Duct propagation (tropospheric ducting): the wave becomes trapped within an atmospheric layer (a 'duct') due to an abnormal refractive index gradient, enabling propagation well beyond normal line-of-sight range.

8

Tropospheric scatter: the wave is scattered by irregularities/turbulence in the troposphere, allowing beyond-line-of-sight communication at VHF/UHF without relying on the ionosphere.

9

The critical frequency is the highest frequency at which a vertically incident wave is reflected back by a given ionospheric layer.

10

Above the critical frequency, the wave penetrates through the ionospheric layer into space rather than being reflected.

11

The ionosphere's ionization density (and hence the critical frequency) varies with time of day, season, and solar activity, directly affecting HF sky-wave propagation range.

12

Free space propagation is an idealized model assuming no obstructions, reflections, or atmospheric effects; received power follows the inverse-square law with distance (Friis transmission equation).

13

Plane earth propagation accounts for a ground-reflected wave in addition to the direct wave, modeling propagation over a flat, reflecting Earth surface.

14

In plane earth propagation, the received signal results from interference between direct and reflected paths.