Maxwell's Equations (Unified)

Equivalent forms

c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}

\Box A^\mu = \mu_0 J^\mu

\partial_\mu F^{\mu\nu} = \mu_0 J^\nu

The same constants you measure with a capacitor and a coil fix the speed of light — electricity and magnetism were never separate, and neither is light.

Symbol	Name	SI unit	Dimension	Range
$c$	Speed of lightoutput Speed of EM waves in vacuum	m/s	L·T⁻¹	290000000 – 310000000
$ε₀$	Vacuum permittivity Electric constant	F/m	M⁻¹·L⁻³·T⁴·I²	8.8e-12 – 8.9e-12
$µ₀$	Vacuum permeability Magnetic constant	H/m	M·L·T⁻²·I⁻²	0.00000125 – 0.00000126

Unit systems

Symbol	SI	CGS	Imperial
$c$	m/s	cm/s	m/s
$ε₀$	F/m	-	F/m
$µ₀$	H/m	-	H/m

Where it holds

Exact classically for all macroscopic fields. Quantum effects (photon statistics, vacuum polarization) require QED; in dense matter use the macroscopic D and H forms with

\varepsilon

, µ.

Dimensional analysis

$1/\surd ([$ µ $_{0}][\varepsilon _{0}]) = 1/\surd ((H/m)(F/m)) = 1/\sqrt{s^{2}/m^{2}} = m/s \checkmark$

Discovery

James Clerk Maxwell · 1865

Maxwell added the 'displacement current' term µ₀ε₀∂E/∂t to Ampère's law to keep charge conserved. The repaired equations supported a self-propagating wave whose speed, computed from ε₀ and µ₀, matched the measured speed of light — unifying optics with electromagnetism.

Try this

Four lines on a T-shirt explain light, radio, magnets, and electricity — and predict the speed of light from two lab constants. How?

Using only ε₀ and µ₀ measured in electrostatics and magnetostatics experiments, show that electromagnetic waves must travel at c = 1/√(µ₀ε₀) ≈ 3×10⁸ m/s.

Research status: stable

Real-world applications

All radio, Wi-Fi, radar, and optical communication
Antenna and waveguide design
Electromagnetic compatibility (EMC) engineering
Foundation for special relativity

Common misconceptions

E and B fields need charges to exist — in a wave they sustain each other in empty space
Maxwell discovered all four — he unified and corrected pre-existing laws, adding the displacement current
The equations are four separate facts — they're one relativistically covariant statement $\partial$ _µF^{µ $\nu }$ = µ $_{0}J^\nu$

Experimental verification

Hertz (1887) generated and detected radio waves, measured their wavelength and frequency, and found speed

= c

— confirming Maxwell's prediction. Today

\varepsilon _{0} and

µ

_{0} are

defined constants and c is exact at 299792458 m/s.

Derivation

In vacuum take the curl of Faraday's law:

\nabla \times (\nabla \times E) = -\partial

ₜ

(\nabla \times B)

.

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

(since

\nabla \cdot E = 0)

and \nabla \times B

= µ

_{0}\varepsilon _{0}\partial

ₜE.

Result:

\nabla ^{2}E

= µ

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

, a wave equation with speed

c = 1/\surd

(µ

_{0}\varepsilon _{0})

.

The same holds for B.

Limiting cases

\partial /\partial t = 0

⟶ electrostatics + magnetostaticsWith no time variation the equations split into Coulomb/Gauss and Ampère/Biot-Savart — E and B decouple.

\rho = 0

,

J = 0

⟶ vacuum wave equationIn empty space each field obeys

\nabla ^{2}F

= µ

_{0}\varepsilon _{0} \partial ^{2}F/\partial t^{2}

— light.

medium with

\varepsilon

, µ⟶

v = 1/\surd

(µ

\varepsilon ) = c/n

In matter the wave slows to c/n;

n = \surd (\varepsilon _r

µ_r).

What if…

What if there were magnetic monopoles?

$\nabla \cdot B$ would equal µ $_{0}\rho _m$ and Faraday's law would gain a magnetic-current term — the equations become beautifully symmetric (see the Dirac monopole card).

What if Maxwell omitted the displacement current?

Ampère's law would violate charge conservation for a charging capacitor, and no EM waves would exist — no light from the theory.

What if you enter a medium with

\varepsilon _r = 4

?

The wave slows to $c/2 (n = 2)$ ; this refraction is the basis of lenses and optical fibers.

1

Speed of light from constants

Given ·

µ₀:: 0.00000125663706212
ε₀:: 8.8541878128e-12

Find · c

Steps

Product µ $_{0}\varepsilon _{0} = 1.2566\times 10^{-6} \times 8.854\times 10^{-12} \approx 1.1127\times 10^{-17} s^{2}/m^{2}$
$\surd = 3.336\times 10^{-9} s/m$
$c = 1/3.336\times 10^{-9} \approx 2.998\times 10^{8} m/s$

Result ·

c = 1/\sqrt{1.2566\times 10^{-6} \times 8.854\times 10^{-12}} \approx 1/\sqrt{1.1127\times 10^{-17}} \approx 2.998\times 10^{8} m/s

Ampère–Maxwell Law→Faraday's Law of Induction→Gauss's Law→Gauss's Law for Magnetism→Speed of Electromagnetic Waves→Poynting Vector→