Ampère–Maxwell Law

Equivalent forms

\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}

I_d = \varepsilon_0 \frac{d\Phi_E}{dt}

One added term completes the symmetry — changing B makes E (Faraday), changing E makes B (Maxwell) — and out falls light. The most consequential patch in the history of physics.

Symbol	Name	SI unit	Dimension	Range
$B$	Magnetic fieldoutput Field magnitude on the Amperian loop	T	M·T⁻²·I⁻¹	0 – 0.0001
$I$	Conduction current Current of moving charges threading the loop	A	I	0 – 10
$dΦᴇ/dt$	Electric flux rate Rate of change of electric flux through the loop's surface	V·m/s	M·L³·T⁻⁴·I⁻¹	0 – 1000000000000
$r$	Loop radius Radius of the circular Amperian loop inside the gap	m	L	0.001 – 0.05

Unit systems

Symbol	SI	CGS	Imperial
$B$	T	G	T
$I$	A	statA	A
$dΦᴇ/dt$	V·m/s	statV·cm/s	V·m/s
$r$	m	cm	in

Where it holds

Exact — one of the four Maxwell equations, valid from statics to optical frequencies and fully consistent with special relativity. In materials, replace

\mu _{0}

,

\varepsilon _{0}

with

\mu

,

\varepsilon and add

bound-current terms (use the H-field form).

Dimensional analysis

$[\varepsilon _{0} d\Phi$ ᴇ/ $dt] = (F/m)(V\cdot m/s)$ = $(A\cdot s/V\cdot m)(V\cdot m/s) = A$ — the displacement term really is a current $\checkmark$

Discovery

James Clerk Maxwell · 1861

Maxwell added the displacement-current term on largely theoretical grounds in 'On Physical Lines of Force' (1861) — the single correction that made the equations consistent with charge conservation and predicted electromagnetic waves. Hertz confirmed the waves in 1887, eight years after Maxwell's death.

Try this

A charging capacitor has NO current between its plates — so why is there a magnetic field circling in the empty gap?

A 2 A current charges a capacitor with circular plates of radius 5 cm. Find the magnetic field 2.5 cm from the axis, inside the gap where no charge flows.

Research status: stable

Real-world applications

Radio, Wi-Fi, and all wireless transmission — antennas work because changing E makes B and vice versa
Capacitor behavior at high frequency (displacement current dominates)
Waveguide and microwave cavity design
Medical MRI RF coil field calculations

Common misconceptions

Displacement current is not a flow of charge — nothing moves in the vacuum gap; it is a changing E-field with the magnetic effect of a current
The term is not negligible bookkeeping: without it there are no electromagnetic waves at all
B is continuous across the capacitor region — Ampère's law with either surface (through wire or through gap) now gives the same answer

Experimental verification

Hertz's 1887 spark-gap experiments generated and detected the electromagnetic waves the term predicts — the decisive test. Direct detection of the magnetic field inside a charging capacitor (e.g., Bartlett & Corle, 1985) matches the displacement-current prediction.

Derivation

Take the divergence of Ampère's original

law \nabla \times B = \mu _{0}J

: the left side vanishes identically, forcing

\nabla \cdot J = 0

— which is false when charge accumulates

(\nabla \cdot J = -\partial \rho /\partial t)

.

Substituting

\rho = \varepsilon _{0}\nabla \cdot E

(Gauss) shows the deficit is exactly

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

.

Adding

\mu _{0}\varepsilon _{0} \partial E/\partial t

restores charge conservation identically.

Limiting cases

Static fields

(d\Phi

ᴇ/

dt = 0)

⟶

\oint B\cdot dl = \mu _{0}I

The law reduces to ordinary Ampère's law — magnetostatics is the special case of nothing changing.

Inside the gap, r < R⟶

B = \mu _{0}I r / (2\pi R^{2})

Only the fraction

r^{2}/R^{2} of

the displacement current threads the loop, so B grows linearly with r.

Outside the gap,

r \ge R

⟶

B = \mu _{0}I/(2\pi r)

The full displacement current is enclosed — the field is indistinguishable from that of the wire. No seam in B anywhere.

What if…

What if Maxwell had not added the displacement term?

Charge conservation would be violated and the equations would have no wave solutions — no prediction of light, radio, or any electromagnetic radiation.

What if you choose a surface that bulges through the capacitor gap instead of cutting the wire?

You get the same $\oint B\cdot dl$ : the gap surface intercepts displacement current exactly equal to the wire's conduction current. The law is surface-independent — that was the whole point.

What if the capacitor charges sinusoidally at high frequency?

The oscillating displacement current radiates: the capacitor gap becomes a small antenna. At microwave frequencies, capacitors stop being capacitors.

1

Field inside a charging capacitor

Given ·

I:: 2
r:: 0.025
R:: 0.05

Find · B at

r = 2.5\,\mathrm{cm}

Steps

Uniform E between plates ⇒ displacement current density is uniform: the loop encloses I_d, $enc = I(r^{2}/R^{2})$
$\oint B\cdot dl = B(2\pi r) = \mu _{0} I r^{2}/R^{2}$
$B = \mu _{0} I r/(2\pi R^{2}) = (1.2566\times 10^{-6} \times 2 \times 0.025)/(2\pi \times 0.0025)$
$B = 4.0\times 10^{-6} T = 4.0 \mu T$ — half the value just outside the plates' edge

Result ·

B = \mu _{0}Ir/(2\pi R^{2}) = 4.0 \mu T

2

Displacement current equals conduction current

Given ·

I:: 2
R:: 0.05

Find · dE/dt between the plates

Steps

Total displacement current must equal the wire current: $I = \varepsilon _{0} (dE/dt) \pi R^{2}$
$dE/dt = 2/(8.854\times 10^{-12} \times \pi \times 0.0025)$
$\approx 2.88\times 10^{13} V/(m\cdot s)$ — huge field rates from a modest 2 A, because $\varepsilon _{0} is$ tiny

Result ·

dE/dt = I/(\varepsilon _{0}\pi R^{2}) \approx 2.88\times 10^{13} V/(m\cdot s)

Ampère's Law (with Maxwell's correction)→Faraday's Law of Induction→Speed of Electromagnetic Waves→Gauss's Law→