Poynting Vector

Definition

The Poynting vector in SI units is:

S = E × H = (1/μ₀)·E × B    [W/m²]

It is a flux density — the rate of energy transport per unit area perpendicular to S. Direction follows the right-hand rule applied to E and H.

In Gaussian units S = (c/4π)·E × B; the dimensions still come out as energy/area/time.

Poynting's theorem

Start from Maxwell's equations. Multiply Faraday's law by H, Ampère's law by E, subtract, and rearrange. You get:

∂u/∂t + ∇·S = −J·E

u = ½·(ε₀·E² + B²/μ₀)   [energy density of the EM field, J/m³]
S = E × H                [energy flux density, W/m²]
J·E                       [power per unit volume done by fields on charges]

Integrate over a volume V bounded by surface ∂V:

d/dt ∫_V u dV + ∮_{∂V} S · dA = −∫_V J·E dV

Translation: rate of change of EM energy inside + power flowing out through the boundary = power dissipated into matter inside. Energy conservation, expressed entirely in field language.

Plane wave example

Consider a linearly polarized plane wave traveling in +z, with E pointing along x and B along y:

E(z, t) = E₀·cos(kz − ωt)·x̂
B(z, t) = (E₀/c)·cos(kz − ωt)·ŷ
S = E × H = (E₀²/(μ₀·c))·cos²(kz − ωt)·ẑ

Time average:
⟨S⟩ = (E₀²)/(2·μ₀·c) = (½)·ε₀·c·E₀² = ½·c·ε₀·E₀²    [W/m²]

The time-averaged intensity is half the peak — same factor of ½ you see in AC power. In a non-magnetic medium with refractive index n, replace c with c/n and add factor n.

Real-world magnitudes

Source	|⟨S⟩|	Peak E-field
Solar constant (top of atmosphere)	1361 W/m²	~1000 V/m
Sunlight at Earth's surface (clear noon)	≈ 1000 W/m²	~870 V/m
Sun's photosphere (Stefan-Boltzmann at 5778 K)	6.32 × 10⁷ W/m²	~2.2 × 10⁵ V/m
Microwave oven interior	~10⁴ W/m²	~2700 V/m
Smartphone WiFi (1 m away)	~10⁻⁴ W/m²	~0.3 V/m
FM radio at 10 km from 50 kW transmitter	~4 × 10⁻⁶ W/m²	~0.05 V/m
Cell-phone-tower exposure limit (FCC)	10 W/m² at 1900 MHz	~87 V/m
CMB background	3.13 × 10⁻⁶ W/m²	tiny
Pulsed laser (1 mJ, 100 fs, 100 μm² spot)	~10¹⁸ W/m²	~10⁹ V/m

Numerical sanity check: ⟨S⟩ = 1000 W/m² ⟹ E₀ = √(2·μ₀·c·⟨S⟩) = √(2 · 4π·10⁻⁷ · 3×10⁸ · 1000) ≈ 870 V/m. So peak sunlight E-field at Earth's surface is roughly 1 kV per meter — surprising once you see it written down.

Worked example — radiated power from a dipole antenna

A short electric dipole of length L driven at angular frequency ω with current I₀ radiates in a doughnut pattern. The far-field Poynting magnitude on the equatorial plane at distance r is:

⟨|S|⟩ = (μ₀·ω²·I₀²·L²)/(32·π²·c·r²) · sin²θ

Total radiated power (integrate over the sphere):
P_rad = (μ₀·ω²·I₀²·L²)/(12·π·c) = ⅔ · (μ₀·c/12π) · (ω·I₀·L/c)²

For a 1 m dipole carrying 1 A at 100 MHz (ω = 6.28 × 10⁸ rad/s):
P_rad ≈ (4π·10⁻⁷ · (6.28×10⁸)² · 1² · 1²) / (12π·3×10⁸)
       ≈ 4.4 W

At r = 100 m, equatorial Poynting:
⟨|S|⟩ ≈ 4.4 / (4π · 100²) ≈ 3.5 × 10⁻⁵ W/m²

That is roughly the same magnitude as cellphone-tower exposure at moderate range, illustrating why exposure-limit calculations are essentially Poynting-vector accounting.

Radiation pressure

The momentum density carried by the EM field is g = S/c². A wave absorbed by a perfectly black surface exerts pressure P_rad = ⟨|S|⟩/c; a perfect reflector doubles it (2⟨|S|⟩/c). For sunlight at Earth (1361 W/m²):

P_rad(absorbed) = 1361 / (3 × 10⁸) ≈ 4.5 × 10⁻⁶ Pa
P_rad(reflected) ≈ 9 × 10⁻⁶ Pa

Tiny — but enough to accelerate solar-sail spacecraft like LightSail-2 over months, and to alter the orbits of small asteroids (Yarkovsky and YORP effects).

JavaScript — Poynting-vector arithmetic

const mu0 = 4 * Math.PI * 1e-7;   // T·m/A
const eps0 = 8.854e-12;           // F/m
const c = 2.998e8;                // m/s

// Time-averaged Poynting magnitude for a plane wave
function S_avg(E0) { return (E0 * E0) / (2 * mu0 * c); }

// Peak E-field from time-averaged intensity
function E_from_S(S) { return Math.sqrt(2 * mu0 * c * S); }

console.log(`Solar S ≈ 1361 W/m² → E ≈ ${E_from_S(1361).toFixed(0)} V/m`);  // ~1011

// Radiation pressure
function radiationPressure(S, mode = 'absorb') {
  return mode === 'reflect' ? 2 * S / c : S / c;
}
console.log(`Solar sail pressure (reflect): ${radiationPressure(1361, 'reflect').toExponential(2)} Pa`);

// Total radiated power from short dipole
function dipoleRadiatedPower(I0, L, freq) {
  const omega = 2 * Math.PI * freq;
  return (mu0 * omega * omega * I0 * I0 * L * L) / (12 * Math.PI * c);
}
console.log(`1 A, 1 m, 100 MHz dipole: P_rad ≈ ${dipoleRadiatedPower(1, 1, 1e8).toFixed(1)} W`);

// Equatorial far-field Poynting at distance r
function dipoleS_equatorial(I0, L, freq, r) {
  const omega = 2 * Math.PI * freq;
  return (mu0 * omega * omega * I0 * I0 * L * L) / (32 * Math.PI * Math.PI * c * r * r);
}
console.log(`Dipole at 100 m: ${dipoleS_equatorial(1, 1, 1e8, 100).toExponential(2)} W/m²`);

// Solar irradiance per area of panel
function panelPower(area_m2, eff = 0.20, intensity = 1000) {
  return area_m2 * intensity * eff;
}
console.log(`2 m² panel at 1000 W/m², 20%: ${panelPower(2, 0.20, 1000)} W`);  // 400 W

// Stefan-Boltzmann surface flux for blackbody at T
function blackbodyFlux(T) {
  const sigma = 5.670e-8;
  return sigma * Math.pow(T, 4);
}
console.log(`Sun surface flux at 5778 K: ${blackbodyFlux(5778).toExponential(2)} W/m²`);  // ~6.3e7

Where the Poynting vector shows up

• Antenna engineering. Directivity, gain, beam pattern, effective aperture — all defined via the angular distribution of ⟨|S|⟩ in the far field.
• Solar power and photovoltaics. Module rating is W per m² × area × efficiency. The 1361 W/m² solar constant is a Poynting magnitude.
• Optics. Light intensity, Beer-Lambert absorption, refractive-index interfaces — all written in terms of S. Reflection and transmission coefficients are ratios of Poynting fluxes.
• Transformer and motor design. Real magnetic-circuit engineers use Poynting flux to track where the power physically flows; counter-intuitively, transformer power flows through the air gap, not the iron.
• RF safety and exposure. FCC, ICNIRP, and other regulators specify maximum permissible exposure in W/m² (or equivalent V/m) — direct Poynting-vector limits.
• Wireless power transfer. Inductive charging, beamed microwave power (Apple/Qi, future space-based solar). Efficiency is measured as Poynting flux delivered vs. transmitted.
• Astrophysics. Stellar luminosities (W), pulsar spin-down power, jet kinetic luminosity — all integrated Poynting fluxes over the stellar surface or magnetosphere.
• Solar sails and radiation pressure. Momentum carried by the EM field equals S/c²; that's the underlying mechanism of every photon-pushed spacecraft.

Common mistakes

• Confusing E × H with E × B. In SI units S = E × H = (1/μ₀)·E × B. The factor of μ₀ matters: using E × B without dividing gives the wrong units and a 10⁷-fold error.
• Forgetting the time average. Instantaneous |E × H| oscillates at 2ω; for power calculations you want ⟨|S|⟩ = (1/2)·|E₀ × H₀|. Skipping the factor of ½ doubles your answer.
• Thinking energy flows through wires in a DC circuit. S = E × H around a current-carrying wire points sideways from the surrounding space into the resistor — energy travels through the field, not through the metal. The metal just guides the field.
• Treating S as gauge-invariant in detail. Total radiated power (∮S·dA) is unambiguous, but the local definition allows adding curls of arbitrary fields without changing the conservation law. The standard choice S = E × H is canonical because it correctly tracks momentum too.
• Ignoring the medium. In materials with refractive index n, the EM energy moves at c/n, and the Poynting magnitude needs μ replaced by μ = μ₀μ_r. For non-magnetic dielectrics S ≈ n·ε₀·c·⟨E²⟩.
• Forgetting that S can point into a conductor. Even though the perfect conductor has no internal S, real conductors with finite skin depth absorb a thin sheet of Poynting flux — that's how ohmic heating gets accounted for in field language.