Intensity of an Electromagnetic Wave

Equivalent forms

I = \frac{c \epsilon_0}{2} E_0^2

I = \frac{E_0^2}{2 \mu_0 c}

\langle S \rangle = \frac{1}{2} c \epsilon_0 E_0^2

Power flow of every electromagnetic wave — from radio to gamma — fits into a single product of c, epsilon_0, and amplitude squared.

Symbol	Name	SI unit	Dimension	Range
$c$	Speed of light Speed of light in vacuum	m/s	L T^-1	299792458 – 299792458
$epsilon_0$	Vacuum permittivity Electric constant	F/m	M^-1 L^-3 T^4 I^2	8.854e-12 – 8.854e-12
$E_0$	Electric field amplitude Peak amplitude of the electric field	V/m	M L T^-3 I^-1	0 – 10000
$I$	Intensityoutput Time-averaged power per unit area	W/m^2	M T^-3	0 – 10000000

Unit systems

Symbol	SI	CGS	Imperial
$E_0$	V/m	statV/cm	V/m
$I$	W/m^2	erg/(s*cm^2)	W/m^2

Where it holds

Valid for monochromatic plane waves in vacuum or low-loss dielectrics. In conducting or absorbing media, additional attenuation terms apply.

Discovery

John Henry Poynting · 1884

Poynting derived the energy flow vector S = (1/mu_0) E x B from Maxwell's equations. Time-averaging the oscillating product gives the practical intensity formula used throughout optics and radio engineering.

Try this

How much power does sunlight actually deliver per square meter?

A monochromatic laser has peak electric field amplitude E_0 = 1000 V/m. Find its time-averaged intensity.

Research status: stable

Real-world applications

Solar panel design: solar irradiance $\sim 1361\,\mathrm{W/m}^2$ sets maximum output.
Laser safety thresholds: maximum permissible exposure limits in W/cm^2.
Radio link budget: receiver field strength scales with sqrt(intensity).
Radiation pressure calculations for solar sails and stellar light pressure.

Common misconceptions

Intensity is linear in E — it is quadratic, like any energy density.
Field amplitude and intensity are interchangeable — they have different units and scaling.
The 1/2 factor is from polarization — it comes from time-averaging cos^2.

Experimental verification

Verified by radiometers, photodiodes, and bolometers calibrated against standard sources. Solar constant

(\sim 1361\,\mathrm{W/m}^2

above atmosphere) is a direct measurement.

Derivation

The instantaneous Poynting vector is

S = (1/\mu _0) E x B

.

For a plane wave,

B = E/c

, so |

S| = E^2/(\mu _0\,\mathrm{c})

.

Substituting

E = E_0 \cos (\omega t)

and time-averaging gives <S>

= E_0^2/(2 \mu _0\,\mathrm{c}) = (1/2) c \varepsilon _0\,\mathrm{E}_0^2

.

Limiting cases

E_0 \to 0

⟶

I \to 0

No field, no energy flow.

E_0 doubles⟶ I quadruplesQuadratic scaling — typical of energy in any oscillator.

E_0 \sim 3e_{6} V/m

⟶

I \sim 1.2e10\,\mathrm{W/m}^2

Air breakdown threshold — sparks form, plasma generation begins.

What if…

What if you use the RMS field instead of peak?

$I = c*\varepsilon _0*E_rms^2$ (no 1/2), since $E_rms = E_0/\sqrt{2}$ .

What if the wave is in a medium with refractive index n?

Replace epsilon_0 by n^2*epsilon_0 and c by c/n; intensity formula becomes $I = (1/2) n c \varepsilon _0\,\mathrm{E}_0^2$ .

1

Intensity of a laser with 1000 V/m amplitude

Given ·

c:: 299792458
epsilon 0:: 8.854e-12
E 0:: 1000

Find · I

Steps

Apply intensity formula: $I = (1/2) * c * \varepsilon _0 * E_0^2$
Compute c*epsilon_0: $299792458 * 8.854e-12 = 2.654e-3$
Multiply by E_0^2: $2.654e-3 * 1e_{6} = 2654$
Halve: $I = 1327\,\mathrm{W/m}^2$ (comparable to solar surface intensity at Earth)

Result ·

I = 0.5 * 299792458 * 8.854e-12 * 1000^2 = 0.5 * 2.654e-3 * 1e_{6} = 1327\,\mathrm{W/m}^2

Wave Speed Equation→Refractive Index Definition→Malus's Law→