Speed of Electromagnetic Waves

Equivalent forms

v = \frac{c}{n} = \frac{1}{\sqrt{\mu \varepsilon}}

c = \frac{E_0}{B_0}

Two tabletop constants — one from capacitors, one from coils — multiply to give the speed of light. Arguably the greatest unification in physics before relativity, and the seed of relativity itself.

Symbol	Name	SI unit	Dimension	Range
$c$	Wave speedoutput Propagation speed of electromagnetic waves in vacuum	m/s	L·T⁻¹	100000000 – 300000000
$μ₀$	Vacuum permeability Magnetic constant: how strongly currents make magnetic fields	H/m	M·L·T⁻²·I⁻²	0.000001 – 0.000002
$ε₀$	Vacuum permittivity Electric constant: how strongly charges make electric fields	F/m	M⁻¹·L⁻³·T⁴·I²	8e-12 – 9e-12
$n$	Refractive index Factor by which a medium slows the wave: v = c/n	—	1	1 – 2.5

Unit systems

Symbol	SI	CGS	Imperial
$c$	m/s	cm/s	ft/s
$μ₀$	H/m	—	H/m
$ε₀$	F/m	—	F/m
$n$	—	—	—

Where it holds

Exact in vacuum. In media,

v = c/n

holds for the phase velocity of monochromatic waves in linear, transparent materials; near absorption resonances dispersion makes n frequency-dependent and group velocity ≠ phase velocity.

Dimensional analysis

$[\mu _{0}\varepsilon _{0}] = (H/m)(F/m) = (V\cdot s$ / $A\cdot m)$ $(A\cdot s/V\cdot m) = s^{2}/m^{2}$ , so $1/\sqrt{\mu _{0}\varepsilon _{0}}$ has units $m/s \checkmark$

Discovery

James Clerk Maxwell · 1862

Combining his equations, Maxwell found wave solutions traveling at 1/√(μ₀ε₀) ≈ 3.1×10⁸ m/s — startlingly close to Fizeau's measured speed of light. He wrote that we 'can scarcely avoid the inference that light consists in the transverse undulations of the same medium' — unifying optics with electromagnetism in one line.

Try this

Two constants measured with batteries and coils on a lab bench multiply together to predict the speed of light — coincidence?

Compute 1/√(μ₀ε₀) from the tabulated constants. What do you get, and what did Maxwell conclude when he first did this in 1862?

Research status: stable

Real-world applications

GPS timing — signals travel $\sim 30\,\mathrm{cm}$ per nanosecond, so clock errors map directly to position errors
Fiber-optic design: $n \approx 1.47$ sets $\sim 5 \mu s/km$ latency, a hard constraint for trading networks
Radar and lidar ranging
The 1983 SI definition of the meter

Common misconceptions

Light doesn't need a medium — the fields themselves regenerate each other; the Michelson–Morley experiment buried the aether
In glass, photons aren't 'absorbed and re-emitted with delays' — the wave interferes with the material's polarization response, producing a slower collective phase velocity
c is now a defined constant, not a measured one: measuring 'the speed of light' today actually calibrates your ruler

Experimental verification

Weber and Kohlrausch (1856) measured the ratio of electrostatic to electromagnetic charge units and

got \sim 3.1\times 10^{8} m/s

. Modern verification: cavity resonance and laser interferometry agree so precisely that c was fixed by definition at 299 792 458 m/s in 1983.

Derivation

In vacuum (no charges or currents), Maxwell's equations give

\nabla \times (\nabla \times E) = -\partial (\nabla \times B)/\partial t = -\mu _{0}\varepsilon _{0} \partial ^{2}E/\partial t^{2}

.

Using

\nabla \times (\nabla \times E) = -\nabla ^{2}E

for divergence-free fields:

\nabla ^{2}E = \mu _{0}\varepsilon _{0} \partial ^{2}E/\partial t^{2}

— the wave equation with speed

1/\sqrt{\mu _{0}\varepsilon _{0}}

.

Limiting cases

Vacuum

(n = 1)

⟶

v = c = 2.998\times 10^{8} m/s

The maximum signal speed in the universe — exact by definition since 1983, when the meter was redefined from it.

Dielectric medium

(\varepsilon = \kappa \varepsilon _{0})

⟶

v = c/\surd \kappa (non

-magnetic)Bound charges polarize and partially cancel the wave's field, slowing it:

n = \surd \kappa for non

-magnetic materials.

E and B amplitudes⟶

E_{0} = c B_{0}

In a vacuum plane wave the field amplitudes are locked in this ratio — the same c appearing in a different guise.

What if…

What

if \varepsilon _{0}

were twice as large?

c would drop $by \sqrt{2}$ to about $2.1\times 10^{8} m/s$ — and atomic sizes, chemistry, and the fine-structure constant would all shift; the universe would be unrecognizable.

What if a particle travels faster than c/n inside a medium?

Nothing forbids it (only vacuum c is the limit) — it outruns its own field and emits Cherenkov radiation, the electromagnetic analogue of a sonic boom.

What if you chase a light wave at nearly c?

Maxwell's equations give the same speed c in your frame too — the paradox 16-year-old Einstein chewed on, resolved only by special relativity.

1

Maxwell's calculation

Given ·

μ₀:: 0.00000125663706212
ε₀:: 8.8541878128e-12

Find · c

Steps

$\mu _{0}\varepsilon _{0} = 1.25664\times 10^{-6} \times 8.85419\times 10^{-12} = 1.11265\times 10^{-17} s^{2}/m^{2}$
$\sqrt{\mu _{0}\varepsilon _{0}} = 3.33564\times 10^{-9} s/m$
$c = 1/3.33564\times 10^{-9} = 2.99792\times 10^{8} m/s$ — the speed of light

Result ·

c = 2.9979\times 10^{8} m/s

2

Light in water

Given ·

n:: 1.33

Find · v

Steps

$v = c/n = 2.998\times 10^{8} / 1.33$
$v \approx 2.25\times 10^{8} m/s$
Charged particles exceeding this in water emit Cherenkov radiation — the blue glow of reactor pools

Result ·

v = c/1.33 \approx 2.25\times 10^{8} m/s

Ampère–Maxwell Law→Faraday's Law of Induction→Poynting Vector→