Electromagnetic Waves

Maxwell's prediction

Maxwell's equations (1865) describe how electric and magnetic fields interact:

1. Gauss's law for E: ∇·E = ρ/ε₀ (charges create E).
2. Gauss's law for B: ∇·B = 0 (no magnetic monopoles).
3. Faraday's law: ∇×E = -∂B/∂t (changing B creates E).
4. Ampère-Maxwell law: ∇×B = μ₀·J + μ₀ε₀·∂E/∂t (currents and changing E create B).

In vacuum (no charges, no currents), these reduce to wave equations for E and B with speed c = 1/√(ε₀·μ₀) ≈ 3 × 10⁸ m/s — the speed of light.

Wave properties

Property	Value
Speed in vacuum	c = 299,792,458 m/s (exact)
E and B	Perpendicular to each other AND to propagation direction
Magnitude ratio	|E| = c · |B|
Polarization	Direction of E vector (typically labels)
Energy density	u = ½ε₀E² + B²/(2μ₀) = ε₀·E² (equal contributions)
Power per area (Poynting)	S = (1/μ₀) E × B
Momentum per energy	p = E/c

How EM waves are generated

EM waves come from ACCELERATING charges:

• Antenna — current oscillates back and forth; electrons accelerate; emit radio waves.
• Atomic transitions — electrons changing energy levels emit photons (visible, UV, IR).
• Bremsstrahlung — fast electrons decelerating in matter emit X-rays.
• Synchrotron radiation — charged particles accelerated in circles (in B field) emit broad spectrum.
• Thermal radiation — random thermal motion of charges → blackbody spectrum.

JavaScript — EM wave calculations

const c = 299792458;
const epsilon_0 = 8.854e-12;
const mu_0 = 4 * Math.PI * 1e-7;

// Verify c = 1/√(ε₀·μ₀)
console.log(`1/√(ε₀μ₀) = ${(1/Math.sqrt(epsilon_0 * mu_0)).toFixed(0)} m/s`);  // 299,792,458

// Wavelength from frequency (and vice versa)
function freq_to_wavelength(f) { return c / f; }
function wavelength_to_freq(lam) { return c / lam; }

// FM radio (100 MHz)
console.log(`100 MHz wavelength: ${freq_to_wavelength(100e6).toFixed(2)} m`);  // 3 m

// Visible green (550 nm)
console.log(`550 nm frequency: ${(wavelength_to_freq(550e-9) / 1e12).toFixed(0)} THz`);  // ~545

// X-ray (0.1 nm)
console.log(`0.1 nm freq: ${(wavelength_to_freq(0.1e-9) / 1e18).toFixed(2)} EHz`);

// Photon energy
const h = 6.626e-34;
function photonEnergy(f) { return h * f; }

console.log(`Visible green photon: ${(photonEnergy(545e12) * 6.242e18).toFixed(2)} eV`);  // ~2.25 eV
console.log(`X-ray photon (10 keV): wavelength ${(c / (10000 * 1.602e-19 / h) * 1e9).toFixed(3)} nm`);

// Poynting vector magnitude (intensity)
function poyntingMagnitude(E_peak) {
  // For sinusoidal: S_avg = ε_0 · c · E_peak² / 2
  return epsilon_0 * c * E_peak * E_peak / 2;
}

// Sunlight: 1361 W/m² → E_peak?
const E_sun = Math.sqrt(2 * 1361 / (epsilon_0 * c));
console.log(`Sunlight E_peak: ${E_sun.toFixed(0)} V/m`);  // ~1014

// Photon momentum
function photonMomentum(f) {
  return h * f / c;
}

console.log(`Visible photon p: ${photonMomentum(545e12).toExponential(2)} kg·m/s`);

// Radiation pressure (force per area)
function radiationPressure(intensity) {
  // For perfect absorber: P = I/c. For perfect reflector: P = 2I/c
  return intensity / c;
}

console.log(`Sunlight on absorber: ${radiationPressure(1361).toExponential(2)} Pa`);
// ~ 4.5 × 10⁻⁶ Pa — tiny but real

// Dipole antenna: peak power radiated
function dipoleAntenna(I_peak, length, frequency) {
  // P = (μ_0 · ω² · L² · I²) / (12π · c)
  const omega = 2 * Math.PI * frequency;
  return (mu_0 * omega * omega * length * length * I_peak * I_peak) / (12 * Math.PI * c);
}

// 1 m antenna, 1 A, 30 MHz
console.log(`Antenna power: ${dipoleAntenna(1, 1, 30e6).toFixed(2)} W`);

Where EM waves matter

• Communications. Radio, TV, cell phones, WiFi, Bluetooth, satellite — all EM waves.
• Imaging. X-ray, MRI (which uses RF, not just B), microscopy, telescope.
• Navigation. GPS uses microwave signals from satellites.
• Heating. Microwaves heat food; IR heats from sun, fire.
• Spectroscopy. Identify materials by their EM-wave absorption/emission patterns.
• Lasers. Coherent EM waves for cutting, surgery, fiber optics, communications.
• Astronomy. Light from stars/galaxies tells us composition, motion, distance.

Common mistakes

• Thinking light needs a medium. EM waves propagate through vacuum. No "ether" exists.
• Treating E and B as independent. They're locked together in EM waves. Both required for propagation.
• Confusing speed in matter with c. c is vacuum speed. In matter, v = c/n < c.
• Forgetting EM waves carry momentum. Massless but momentum p = E/c. Push solar sails, comet tails (radiation pressure).
• Confusing intensity with E peak. Intensity I = ½ε₀cE² (time-averaged power per area). E and B fluctuate; intensity is steady (for sinusoidal).
• Assuming all EM waves visible. Visible is a tiny slice (380-750 nm). Most EM waves invisible to humans.