Poynting Vector — Physica

Equivalent forms

\vec{S} = \vec{E} \times \vec{H}

\langle S \rangle = \tfrac{1}{2} c \varepsilon_0 E_0^2

Once you accept S = (1/μ₀)E×B, every observation about light — radiation pressure, momentum, intensity — falls out as a corollary. A single vector unifies the energetics of all EM fields.

Symbol	Name	SI unit	Dimension	Range
$S$	Poynting magnitudeoutput Instantaneous power per unit area	W/m²	M·T⁻³	0 – 10000
$E$	Electric field amplitude Magnitude of the electric field	V/m	M·L·T⁻³·I⁻¹	0 – 10000
$B$	Magnetic flux density Magnitude of the magnetic field (perpendicular to E for a plane wave)	T	M·T⁻²·I⁻¹	0 – 0.001

Unit systems

Symbol	SI	CGS	Imperial
$S$	W/m²	erg/(s·cm²)	W/m²
$E$	V/m	statV/cm	V/m
$B$	T	G	T

Where it holds

Valid universally in classical electrodynamics. In media replace

\mu _{0}

with

\mu

;

S = E \times H

. Quantum-electrodynamic corrections matter only at extreme field strengths near the Schwinger limit

(\sim 10^{18} V/m)

.

Dimensional analysis

$[S] = [E][B]/[\mu _{0}] = (V/m)(T)/(H/m) = (V\cdot T)/(V\cdot s$ /A $) = A\cdot T = A\cdot (kg$ / $(A\cdot s^{2})) = kg\cdot s^{-3} = W/m^{2} \checkmark$

Discovery

John Henry Poynting · 1884

Poynting derived the energy-flux vector from Maxwell's equations, showing that EM energy flows even through empty space surrounding currents — a revolutionary departure from the idea that energy lived only in matter.

Try this

How does electromagnetic energy actually travel through space — and in what direction?

Sunlight at Earth's surface has E ≈ 1000 V/m. Estimate the magnetic field amplitude and the average power per unit area carried by the wave.

Research status: stable

Real-world applications

Solar irradiance measurements at Earth's surface $(1361\,\mathrm{W/m}^{2} at top$ of atmosphere)
Laser power-density calibration
Antenna engineering and far-field beam patterns
Radiation-pressure-driven solar sails

Common misconceptions

Energy in a DC circuit also flows via S — through the field surrounding the wire, not through the wire itself
S is generally not in the direction of current; it is along $E \times B$
Instantaneous S can be negative (energy flowing 'backward') in standing waves

Experimental verification

Radiometers measure radiation pressure

P = S/c

(or 2S/c for perfect reflectors) — Nichols and Hull (1901) directly confirmed Maxwell's prediction. Modern solar-cell calibrations rely on the same identity.

Derivation

Take

\partial u/\partial t

for EM energy density

u = (\varepsilon _{0} E^{2})/2 + B^{2}/(2\mu _{0})

, substitute Maxwell's curl equations, and integrate by parts to

get \partial u/\partial t + \nabla \cdot S = -J\cdot E

.

The vector

S = (1/\mu _{0}) E \times B

emerges as the energy-flux density.

Limiting cases

E parallel to B⟶

S = 0

No energy flows when E and B are parallel — the cross product vanishes.

Plane wave in vacuum⟶

B = E/c

The two fields are locked together;

S = E^{2}/(\mu _{0} c)

.

Time average for sinusoidal wave⟶ ⟨S⟩

= (1/2)c\varepsilon _{0} E_{0}^{2}

The instantaneous S oscillates at

2\omega

; halving the peak gives the rms-style average.

What if…

What if E doubles?

S quadruples — intensity scales with $E^{2}$ .

What if the wave hits a perfectly reflecting mirror?

Radiation pressure doubles to 2S/c — the basis of solar-sail propulsion.

What if you're inside a DC current-carrying resistor?

S still exists — it points radially inward, delivering the $I^{2}R$ dissipation through the surrounding field, not through the wire interior.

1

Sunlight at Earth's surface

Given ·

E:: 1000
c:: 299792458

Find · ⟨S⟩

Steps

Plane wave in vacuum: $B = E/c \approx 1000/3\times 10^{8} \approx 3.33\times 10^{-6} T$
Apply ⟨S⟩ $= (1/2)c\varepsilon _{0} E^{2}$
⟨S⟩ $= 0.5 \times 3\times 10^{8} \times 8.854\times 10^{-12} \times 10^{6}$
⟨S⟩ $\approx 1326\,\mathrm{W/m}^{2}$ — matches the solar constant

Result · ⟨S⟩

\approx 1326\,\mathrm{W/m}^{2}

2

Power through a 1 m² window

Given ·

S:: 800
A:: 1

Find · Total power

Steps

$P =$ ⟨S⟩ $\times A = 800 \times 1 = 800\,\mathrm{W}$
This is roughly the noon irradiance on a clear day, integrated over $1\,\mathrm{m}^{2}$

Result ·

P = 800\,\mathrm{W}

Lorentz Force Law→Ampère's Law (with Maxwell's correction)→Faraday's Law of Induction→Gauss's Law→