Learning #25 – Electromagnetism: Maxwell’s Equations, EM Waves, Antennas and Transmission Lines

James Clerk Maxwell’s 1865 synthesis of electricity and magnetism into a single set of four field equations is one of the greatest theoretical achievements in physics. The equations predict the speed of light from purely electromagnetic constants, unified optics with electricity, and laid the foundation for wireless communication, electrical power systems, and quantum electrodynamics. Every antenna, transformer, optical fibre, and microwave oven operates on principles encoded in Maxwell’s equations.

1. Maxwell’s Equations — The Complete Framework

The four Maxwell equations relate the electric field E, magnetic field B, charge density ρ and current density J. In differential form they apply at every point in space and time; in integral form they relate flux and circulation over surfaces and loops. Together they are a complete classical description of all electromagnetic phenomena.

Differential form (SI, in vacuum or linear medium):

  ∇·E = ρ/ε_0                            (Gauss’s law: E-field diverges from charges)
  ∇·B = 0                                   (no magnetic monopoles: B has no divergence)
  ∇×E = −∂B/∂t                        (Faraday: changing B induces circulating E)
  ∇×B = μ_0(J + ε_0 ∂E/∂t)         (Ampère + Maxwell: currents and changing E generate B)

Integral form:
  &oiint; E·dA = Q_enc/ε_0                   (total flux through closed surface = enclosed charge)
  &oiint; B·dA = 0                               (no isolated magnetic poles)
  ∮ E·dl = −dΦ_B/dt                     (EMF around loop = −rate of change of magnetic flux)
  ∮ B·dl = μ_0(I_enc + ε_0 dΦ_E/dt)   (magnetic circulation = enclosed current + displacement current)

Maxwell’s addition: displacement current ε_0 ∂E/∂t
  Without it, ∇·J = −∂ρ/∂t (charge conservation) is inconsistent with original Ampère.
  Physical meaning: a changing E field in a capacitor gap acts like a current — no charge crosses.

Constants:
  ε_0 = 8.854 × 10^−12 F/m  (permittivity of free space)
  μ_0 = 4π × 10^−7 H/m      (permeability of free space)
  c = 1/√(μ_0ε_0) = 2.998 × 10^8 m/s  (speed of light, predicted by Maxwell 1865)

2. Electromagnetic Waves and the Poynting Vector

In free space (no sources), Maxwell’s equations can be combined to give the wave equation: each component of E and B satisfies (∇² − (1/c²)∂²/∂t²) f = 0. The solutions are transverse electromagnetic (TEM) waves: E and B oscillate perpendicular to each other and to the propagation direction, in phase, with |E|/|B| = c.

Monochromatic plane wave propagating in +z direction:
  E(z,t) = E_0 cos(kz − ωt) x̂
  B(z,t) = (E_0/c) cos(kz − ωt) ŷ
  k = ω/c = 2π/λ   (wave number)

Energy and momentum density:
  u_E = (1/2)ε_0 E²    (electric energy density)
  u_B = (1/2μ_0) B²         (magnetic energy density)
  u_E = u_B for a plane wave (equal partition)
  Total: u = ε_0 E² = B²/μ_0

Poynting vector (energy flux density):
  S = (1/μ_0) E × B   [W/m²]
  |S| = u·c = E²/(μ_0 c)   for a plane wave

Time-averaged intensity (irradiance):
  ⟨S⟩ = E_0² / (2μ_0 c)  [W/m²]
  At Earth’s surface: solar irradiance ~ 1361 W/m²

Radiation pressure:
  P_rad = ⟨S⟩/c   (perfectly absorbing surface)
  P_rad = 2⟨S⟩/c  (perfectly reflecting surface)
  Solar radiation pressure at 1 AU: ~ 9 × 10^−6 Pa (small but significant for spacecraft)

EM spectrum (frequency ranges):
  Radio:       f < 300 MHz   (λ > 1 m)
  Microwave:   300 MHz – 300 GHz  (λ 1 mm – 1 m)
  Infrared:    300 GHz – 430 THz  (λ 700 nm – 1 mm)
  Visible:     430 – 770 THz  (λ 390 – 700 nm)
  UV:          770 THz – 30 PHz  (λ 10 – 390 nm)
  X-ray:       30 PHz – 30 EHz  (λ 0.01 – 10 nm)
  Gamma:       f > 30 EHz    (λ < 0.01 nm)

3. Faraday Induction and Lenz’s Law

Faraday’s law of induction states that a changing magnetic flux through a circuit induces an electromotive force (EMF). Lenz’s law characterises the sign: the induced current always opposes the change that created it (energy conservation in disguise). Together, Faraday induction and Lenz’s law are the operating principles of generators, transformers, induction motors, wireless charging, and metal detectors.

Faraday’s law:
  &EMcycl; = −dΦ_B / dt   where Φ_B = &iint; B·dA (magnetic flux through surface)
  For N-turn coil:  &EMcycl; = −N dΦ_B/dt

Lenz’s law: The minus sign ensures the induced EMF drives a current that opposes the flux change.

Motional EMF:
  A conductor of length L moving at velocity v perpendicular to B:
  &EMcycl; = BLv   (Lorentz force on free charges in the conductor)
  Faraday generator: rotate loop (area A, N turns) at ω:
  &EMcycl;(t) = NBAω sin(ωt)   (sinusoidal AC output)

Self-inductance L:
  Φ_B = LI   (flux through coil proportional to its own current)
  V_L = −L dI/dt   (voltage across inductor opposes current change)
  Energy stored: U_L = (1/2)LI²

Solenoid self-inductance:
  L = μ_0 n² V   (n = turns per meter, V = volume)
  For 1 000-turn, 10 cm radius, 50 cm solenoid: L ≈ 3.9 mH

Transformer (ideal):
  V_1/V_2 = N_1/N_2   (voltage ratio = turns ratio)
  I_1/I_2 = N_2/N_1   (current ratio inverse)
  P_1 = P_2            (conservation of energy, assuming 100% efficiency)
  Real transformers: resistive losses (I²R), eddy current losses, hysteresis losses

4. Antenna Radiation — From Current to Far Field

An antenna converts time-varying electric currents into propagating electromagnetic waves (transmit mode) or vice versa (receive mode). The radiation pattern — the angular distribution of radiated power — follows from the retarded potentials and the acceleration of charges. Even the simplest antenna, the Hertzian dipole, encapsulates the essential physics of all radiation.

Hertzian dipole (infinitesimally short current element I dℓ):
  Electric far field:
    E_θ = (I dℓ / 4π) · (μ_0 ω²/c) · (sinθ/r) · cos(ωt − kr)
  Magnetic far field:
    H_φ = E_θ / η_0   (η_0 = √(μ_0/ε_0) = 377 Ω wave impedance)
  Both fields decay as 1/r (far field); near field has 1/r² and 1/r³ terms.

Radiation pattern:
  Power per unit solid angle: dP/dΩ ∝ sin²θ  (donut shape, null on dipole axis)

Total radiated power (Larmor formula for accelerating charge):
  P = q²a² / (6πε_0 c³)   (a = acceleration, q = charge)
  For a current element: P = (I² dℓ² ω&sup4; μ_0) / (12π c³)  ∝ f&sup4;
  Higher frequencies radiate much more efficiently (why AM needs huge transmitter, FM less so).

Key antenna parameters:
  Directivity D: ratio of main-lobe intensity to isotropic average
    Hertzian dipole: D = 1.5 (1.76 dBi)
    Half-wave dipole: D = 1.64 (2.15 dBi)
    10-element Yagi: D ~ 10–14 dBi
  Gain G = D · η (η = radiation efficiency, < 1 due to ohmic losses)
  Bandwidth: range of f for SWR < 2 (often 3–5% for simple dipoles)
  Impedance: half-wave dipole Z_in ≈ 73 + j42.5 Ω at resonance

Friis transmission equation (link budget):
  P_received = P_transmitted · G_T · G_R · (λ/4πd)²
  Free-space path loss: FSPL = (4πd/λ)²
  At 2.4 GHz over 100 m: FSPL ~ 80 dB (Wi-Fi budget critical)

5. Skin Effect — Why High-Frequency Currents Avoid Volume

When an alternating current flows in a conductor, the induced eddy currents oppose flux penetration, pushing the current toward the surface. The current density decays exponentially with depth, with a characteristic depth called the skin depth δ. At high frequencies δ can be a fraction of a millimetre, meaning that a copper wire at 10 GHz carries current only in a thin shell — its bulk resistance becomes irrelevant. This effect governs PCB trace losses, transformer core selection, and RF shield design.

Wave equation in a conductor (complex permittivity):
  ∇²E = jωμσE   (σ = conductivity)
  Solution: E(x) = E_0 exp(−x/δ) exp(−jx/δ)
  (both decay and phase-shift with depth x)

Skin depth:
  δ = √(2/ωμσ) = 1/√(π f μ σ)
  f ↑ → δ ↓ (current confined to thinner layer)

Values for copper (σ = 5.96 × 10^7 S/m):
  50 Hz:     δ = 9.3 mm   (fully used at power-line frequency for typical cables)
  1 kHz:     δ = 2.1 mm
  1 MHz:     δ = 66 µm
  100 MHz:   δ = 6.6 µm
  10 GHz:    δ = 0.66 µm  (microwave circuits)

Surface resistance (effective resistance per square):
  R_s = 1/(σδ) = √(π f μ / σ)   [Ω/square]
  A 1 GHz copper trace: R_s = 0.026 Ω/square — significant for mm-length traces.

Consequently:
  AC resistance R_AC / R_DC = a/(2δ) for wire radius a ≫ δ
  Litz wire: many fine strands of diameter < 2δ wound together
  → reduces effective resistance at audio and RF frequencies.

6. Transmission Lines — Guided Waves and Standing-Wave Ratio

A transmission line — coaxial cable, microstrip, waveguide — guides electromagnetic waves between source and load. When the load impedance is not matched to the line’s characteristic impedance Z_0, some signal is reflected, creating a standing wave. The standing-wave ratio (SWR) measures the mismatch; maximum power transfer requires SWR = 1 (perfect match). Transmission-line theory is essential for RF design, PCB layout above GHz frequencies, and signal-integrity work.

Telegrapher’s equations (distributed circuit model):
  ∂V/∂x = −(R + jωL) I
  ∂I/∂x = −(G + jωC) V
  Propagation constant: γ = α + jβ = √((R+jωL)(G+jωC))
  Characteristic impedance: Z_0 = √((R+jωL)/(G+jωC))

Lossless transmission line (R=0, G=0):
  Z_0 = √(L/C),  β = ω√(LC),  v_p = 1/√(LC)
  Coaxial cable: Z_0 = (60/√ε_r) ln(b/a) Ω   (b/a = ratio of outer/inner radii)
  Microstrip:    Z_0 depends on w/h ratio and substrate ε_r (computed numerically)

Reflection coefficient at load Z_L:
  Γ = (Z_L − Z_0) / (Z_L + Z_0)   (complex, |Γ| ≤ 1)
  Short circuit: Z_L = 0 → Γ = −1  (total reflection, inverted)
  Open circuit:  Z_L = ∞ → Γ = +1  (total reflection, same phase)
  Matched:       Z_L = Z_0 → Γ = 0  (no reflection)

Voltage standing-wave ratio:
  SWR = (1 + |Γ|) / (1 − |Γ|) ∈ [1, ∞)
  SWR = 1: perfect match. SWR = ∞: total reflection (short or open).
  Power reflected: |Γ|² of incident power.

Quarter-wave transformer (impedance matching):
  A λ/4 section of Z_1 = √(Z_0 Z_L) transforms Z_L to Z_0.
  Bandwidth ~ 20% around design frequency.

Smith chart:
  Complex Γ-plane mapped so that constant-resistance circles are easily plotted.
  Used for impedance matching, stability analysis, and filter design.
  Software tools: ADS, CST, SPICE / AWR Microwave Office.