10.03.04 · em-sr / electrodynamics

Displacement current, the continuity equation, and the complete Maxwell equations in matter

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Anchor (Master): Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Sommerfeld, Electrodynamics (1952), Lectures 10-12

Intuition Beginner

Ampere's law says that an electric current creates a magnetic field. But Maxwell noticed a logical gap: when you charge a capacitor, current flows in the wires but not between the plates. Yet the magnetic field around the gap is the same as if the current continued. What creates the field in the gap?

The answer is the displacement current. Between the capacitor plates, the electric field is increasing as charge builds up. Maxwell proposed that a changing electric field creates a magnetic field, just like a real current. The rate of change of the electric field acts as an effective current — the displacement current:

I_{d} = ϵ_{0} \frac{d Φ _{E}}{d t}

where $Φ_{E}$ is the electric flux through the surface. This was Maxwell's key insight: electric and magnetic fields are not just parallel phenomena — they are dynamically coupled. A changing magnetic field creates an electric field (Faraday's law), and a changing electric field creates a magnetic field (displacement current).

The complete Maxwell equations are four equations that govern all of classical electromagnetism: Gauss's law, Gauss's law for magnetism, Faraday's law, and the Ampere-Maxwell law. Together, they predict the existence of electromagnetic waves travelling at the speed of light.

Visual Beginner

Equation	Name	What it says
$\oint D \cdot d a = Q_{f}$	Gauss	Charges create electric fields
$\oint B \cdot d a = 0$	No monopoles	No magnetic charges
$\oint E \cdot d l = - d Φ_{B} / d t$	Faraday	Changing B creates E
$\oint H \cdot d l = I_{f} + d Φ_{D} / d t$	Ampere-Maxwell	Currents and changing E create B

Worked example Beginner

A parallel-plate capacitor with circular plates of radius $R$ is being charged by a current $I$ . Find the magnetic field at radius $s$ from the axis, between the plates.

Between the plates, there is no conduction current but there is a displacement current. The electric field between the plates is $E = Q / (ϵ_{0} A)$ where $Q$ is the accumulated charge and $A = π R^{2}$ . The displacement current is $I_{d} = ϵ_{0} d (E A) / d t = d Q / d t = I$ . The displacement current equals the conduction current.

Applying Ampere's law (in the Ampere-Maxwell form) to a circular loop of radius $s < R$ between the plates: $B \cdot 2 π s = μ_{0} I_{d} \cdot (s^{2} / R^{2}) = μ_{0} I s^{2} / R^{2}$ . So $B = μ_{0} I s / (2 π R^{2})$ — the same field as if a uniform current $I$ filled the entire cross-section. The magnetic field between the plates is identical to the field around the wire.

Check your understanding Beginner

Formal definition Intermediate+

The Ampere-Maxwell law in differential form.

\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

The term $μ_{0} ϵ_{0} \partial E / \partial t = μ_{0} \partial D / \partial t$ is the displacement current density. In matter, the general form is:

\nabla \times H = J_{f} + \frac{\partial D}{\partial t}

The complete Maxwell equations in matter.

\nabla \cdot D = ρ_{f}, \nabla \cdot B = 0, \nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times H = J_{f} + \frac{\partial D}{\partial t}

with the constitutive relations $D = ϵ_{0} E + P$ and $H = B / μ_{0} - M$ , or for linear media: $D = ϵ E$ and $H = B / μ$ .

The continuity equation. Taking the divergence of the Ampere-Maxwell law: $0 = \nabla \cdot (\nabla \times B) = μ_{0} \nabla \cdot J + μ_{0} ϵ_{0} \frac{\partial}{\partial t} (\nabla \cdot E)$ . Using Gauss's law $\nabla \cdot E = ρ / ϵ_{0}$ : $0 = μ_{0} (\nabla \cdot J + \partial ρ / \partial t)$ . Therefore:

\nabla \cdot J + \frac{\partial ρ}{\partial t} = 0

This is the continuity equation — the local statement of charge conservation. The displacement current is precisely the term needed to make the Ampere-Maxwell law consistent with charge conservation.

In vacuum. Setting $ρ = 0$ , $J = 0$ :

\nabla \cdot E = 0, \nabla \cdot B = 0, \nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times B = μ_{0} ϵ_{0} \frac{\partial E}{\partial t}

Taking the curl of the Faraday equation and substituting the Ampere-Maxwell equation gives the wave equation:

\nabla^{2} E = μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}}, c = \frac{1}{μ _{0} ϵ _{0}}

Key derivation Intermediate+

Derivation (Maxwell's correction to Ampere's law).

Theorem. The displacement current $ϵ_{0} \partial E / \partial t$ is uniquely determined by the requirement that Ampere's law be consistent with the continuity equation.

Proof. The original Ampere's law is $\nabla \times B = μ_{0} J$ . Taking the divergence: $0 = μ_{0} \nabla \cdot J$ . This requires $\nabla \cdot J = 0$ , which is only true for steady currents. For time-dependent situations, the continuity equation gives $\nabla \cdot J = - \partial ρ / \partial t$ .

The correction is to replace $J$ with $J + J_{d}$ where $J_{d}$ is chosen so that $\nabla \cdot (J + J_{d}) = 0$ . From the continuity equation: $\nabla \cdot J = - \partial ρ / \partial t$ . From Gauss's law: $ρ = ϵ_{0} \nabla \cdot E$ . So $\nabla \cdot J = - ϵ_{0} \frac{\partial}{\partial t} (\nabla \cdot E) = - \nabla \cdot (ϵ_{0} \frac{\partial E}{\partial t})$ . Therefore $\nabla \cdot (J + ϵ_{0} \frac{\partial E}{\partial t}) = 0$ . The unique correction (up to a curl-free term that vanishes under the curl) is $J_{d} = ϵ_{0} \partial E / \partial t$ . $□$

Derivation (The electromagnetic wave equation in vacuum). Take the curl of Faraday's law: $\nabla \times (\nabla \times E) = - \partial (\nabla \times B) / \partial t = - μ_{0} ϵ_{0} \partial^{2} E / \partial t^{2}$ . Using $\nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E$ and $\nabla \cdot E = 0$ in vacuum: $\nabla^{2} E = μ_{0} ϵ_{0} \partial^{2} E / \partial t^{2}$ . The wave speed is $v = 1/ μ_{0} ϵ_{0} = 3 \times 1 0^{8}$ m/s $= c$ .

Bridge. The displacement current is the keystone of Maxwell's theory: it completes the Ampere-Maxwell law and makes the set of four equations self-consistent (automatically satisfying charge conservation). The foundational insight is that the displacement current creates a symmetric coupling between $E$ and $B$ : changing $E$ creates $B$ (Ampere-Maxwell) and changing $B$ creates $E$ (Faraday), and this mutual coupling produces self-sustaining electromagnetic waves. The central message is that Maxwell's equations in matter are the same as in vacuum, with the substitution $E \to D$ , $B \to H$ , and $ρ, J \to ρ_{f}, J_{f}$ (free charges and currents only). Putting these together, the complete Maxwell equations predict wave propagation in matter 10.04.03 with speed $v = 1/ μ ϵ$ , energy transport via the Poynting vector 10.03.05, and the full range of electromagnetic phenomena from statics to optics.

Exercises Intermediate+

Exercise 5 (hard, symbolic).

Derive Poynting's theorem (energy conservation) from Maxwell's equations: $- \frac{\partial u}{\partial t} = \nabla \cdot S + J \cdot E$ where $u = \frac{1}{2} (ϵ_{0} E^{2} + B^{2} / μ_{0})$ and $S = E \times B / μ_{0}$ .

Hint

Take $E \cdot (\nabla \times B - μ_{0} ϵ_{0} \partial E / \partial t) = μ_{0} J \cdot E$ and use a vector identity for $E \cdot (\nabla \times B)$ .

Answer

From the Ampere-Maxwell law: $E \cdot (\nabla \times B) = μ_{0} J \cdot E + μ_{0} ϵ_{0} E \cdot \frac{\partial E}{\partial t}$ . Using the identity $\nabla \cdot (E \times B) = B \cdot (\nabla \times E) - E \cdot (\nabla \times B)$ , so $E \cdot (\nabla \times B) = B \cdot (\nabla \times E) - \nabla \cdot (E \times B)$ . From Faraday's law: $\nabla \times E = - \partial B / \partial t$ . So $E \cdot (\nabla \times B) = - B \cdot \frac{\partial B}{\partial t} - \nabla \cdot (E \times B)$ . Substituting:

- B \cdot \frac{\partial B}{\partial t} - \nabla \cdot (E \times B) = μ_{0} J \cdot E + μ_{0} ϵ_{0} E \cdot \frac{\partial E}{\partial t}

Rearranging: $- \frac{\partial}{\partial t} [\frac{1}{2 μ _{0}} B^{2} + \frac{ϵ _{0}}{2} E^{2}] = \nabla \cdot (\frac{E \times B}{μ _{0}}) + J \cdot E$ . This is Poynting's theorem: $- \partial u / \partial t = \nabla \cdot S + J \cdot E$ .

Exercise 6 (hard, symbolic).

Show that Maxwell's equations are invariant under the duality transformation $E \to E cos θ + c B sin θ$ , $c B \to - E sin θ + c B cos θ$ (in vacuum with no sources). What happens if magnetic monopoles exist?

Hint

Check that the transformed fields satisfy the same vacuum Maxwell equations. If monopoles exist, the duality also mixes $ρ_{e} \leftrightarrow ρ_{m}$ and $J_{e} \leftrightarrow J_{m}$ .

Answer

In vacuum ( $ρ = 0$ , $J = 0$ ), define $E^{'} = E cos θ + c B sin θ$ and $c B^{'} = - E sin θ + c B cos θ$ .

Gauss: $\nabla \cdot E^{'} = cos θ (\nabla \cdot E) + c sin θ (\nabla \cdot B) = 0 + 0 = 0$ . No monopoles: $\nabla \cdot (c B^{'}) = - sin θ (\nabla \cdot E) + c cos θ (\nabla \cdot B) = 0$ . Faraday: $\nabla \times E^{'} = cos θ (- \partial B / \partial t) + c sin θ (\nabla \times B) / c = - \partial B^{'} / \partial t$ (after substituting the Ampere-Maxwell law). Similarly for Ampere-Maxwell. The equations are invariant.

If magnetic monopoles exist, $\nabla \cdot B = μ_{0} ρ_{m} \neq = 0$ , and the duality transformation mixes electric and magnetic charges: $ρ_{e}^{'} = ρ_{e} cos θ + c ρ_{m} sin θ$ , $c ρ_{m}^{'} = - ρ_{e} sin θ + c ρ_{m} cos θ$ . This means a purely electric charge can be transformed into a purely magnetic charge by choosing $θ = π /2$ . The duality suggests that electric and magnetic charges are fundamentally the same type of object, related by a rotation in "charge space."

Exercise 7 (medium, symbolic).

Write Maxwell's equations in differential-forms notation on 4-dimensional spacetime. Define the 2-form $F$ and identify which equations correspond to $d F = 0$ and $d * F = J$ .

Hint

$F = E_{i} d t \land d x^{i} + \frac{1}{2} B_{i} ϵ_{ij k} d x^{j} \land d x^{k}$ . The homogeneous equations (Gauss B, Faraday) are $d F = 0$ . The inhomogeneous equations (Gauss E, Ampere-Maxwell) are $d * F = J$ .

Answer

The electromagnetic field 2-form is $F = E_{x} d t \land d x + E_{y} d t \land d y + E_{z} d t \land d z + B_{x} d y \land d z + B_{y} d z \land d x + B_{z} d x \land d y$ .

Homogeneous equations $d F = 0$ : This expands to two equations. The $d t$ -components: $\nabla \cdot B = 0$ (Gauss for magnetism). The spatial 3-form components: $\nabla \times E + \partial B / \partial t = 0$ (Faraday's law).

Inhomogeneous equations $d * F = J$ : The Hodge dual $\astF$ swaps $E$ and $B$ (with appropriate signs). The current 3-form is $J = ρ d x \land d y \land d z - J_{x} d t \land d y \land d z - J_{y} d t \land d z \land d x - J_{z} d t \land d x \land d y$ . Expanding $d * F = J$ : $\nabla \cdot E = ρ / ϵ_{0}$ (Gauss) and $\nabla \times B - μ_{0} ϵ_{0} \partial E / \partial t = μ_{0} J$ (Ampere-Maxwell).

The differential-forms notation unifies the four Maxwell equations into two geometric statements: $d F = 0$ (the field is a closed form) and $d * F = J$ (the dual field is sourced by current).

Exercise 8 (medium, symbolic).

The electromagnetic wave speed in a linear medium with permittivity $ϵ$ and permeability $μ$ is $v = 1/ μ ϵ$ . Compute $v$ in glass ( $ϵ_{r} = 4$ , $μ_{r} = 1$ ) and in water ( $ϵ_{r} \approx 80$ , $μ_{r} = 1$ ).

Hint

$v = c / ϵ_{r} μ_{r} = c / n$ where $n$ is the refractive index.

Answer

In glass: $v = c / 4 \times 1 = c /2$ . Light travels at half the speed of light in glass. The refractive index is $n = 2$ .

In water: $v = c / 80 = c /8.94$ . The refractive index would be $n = 8.94$ . However, this uses the static (DC) permittivity. At optical frequencies, the permittivity of water is much lower ( $ϵ_{r} \approx 1.77$ at visible wavelengths), giving $n \approx 1.33$ — the familiar refractive index of water. The DC and optical permittivities differ because the molecular polarisation mechanism cannot respond at optical frequencies (dispersion, covered in 10.04.03).

Lean formalization Intermediate+

Mathlib has vector calculus, differential forms, and the wave equation but not Maxwell's equations, the displacement current, the continuity equation as a consequence, or the constitutive relations. The physical content (charge conservation, the link between changing E and B) is not formalisable without physical axioms. lean_status: none.

Advanced results Master

Potentials and gauge freedom. The complete Maxwell equations in the Lorenz gauge give the wave equations $□ V = - ρ / ϵ_{0}$ and $□ A = - μ_{0} J$ . These are the fundamental equations for computing electromagnetic fields from sources. The retarded solutions are $V (r, t) = \frac{1}{4 π ϵ _{0}} \int \frac{ρ ( r ^{'} , t _{r} )}{∣ r - r ^{'} ∣} d^{3} r^{'}$ and similarly for $A$ , where $t_{r} = t - ∣ r - r^{'} ∣/ c$ is the retarded time.

Hertz vectors. In source-free regions, the electromagnetic field can be expressed in terms of two scalar potentials (the electric and magnetic Hertz vectors $Π_{e}$ and $Π_{m}$ ), each satisfying the wave equation. This decomposition separates the field into TE and TM components, which is useful for waveguide 10.04.04 and cavity 10.04.05 problems.

Maxwell's equations in the language of differential forms. In 4-dimensional notation, Maxwell's equations reduce to two equations: $d F = 0$ (homogeneous) and $d * F = J$ (inhomogeneous). The gauge freedom $F \to F + d A$ (where $A$ is the 4-potential) automatically satisfies the homogeneous equation. The inhomogeneous equation becomes $d * d A = J$ in the Lorenz gauge $d * A = 0$ . This formulation is manifestly Lorentz-invariant and coordinate-free.

Synthesis. The displacement current is the single addition that transforms the pre-Maxwell equations (which described statics and quasi-statics) into the complete Maxwell equations (which describe all electromagnetic phenomena). The foundational insight is that charge conservation (the continuity equation) requires the displacement current: without it, the divergence of the Ampere-Maxwell law would violate $\nabla \cdot (\nabla \times B) = 0$ . The central message is that Maxwell's equations are self-consistent, covariant, and complete: they predict electromagnetic waves 10.04.02, energy conservation 10.03.05, and the entire framework of classical electrodynamics. Putting these together, the displacement current is not merely a correction to Ampere's law but the dynamical term that couples the electric and magnetic fields into a unified electromagnetic field, described covariantly by the Faraday tensor 10.06.01 and supporting radiation 10.07.02.

Full proof set Master

Proposition (The wave equation from Maxwell's equations). In vacuum with $ρ = 0$ and $J = 0$ , both $E$ and $B$ satisfy the wave equation with speed $c = 1/ μ_{0} ϵ_{0}$ .

Proof. Take the curl of Faraday's law: $\nabla \times (\nabla \times E) = - \frac{\partial}{\partial t} (\nabla \times B)$ . Use $\nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E$ . Since $\nabla \cdot E = 0$ in vacuum: $- \nabla^{2} E = - \frac{\partial}{\partial t} (μ_{0} ϵ_{0} \frac{\partial E}{\partial t})$ . So $\nabla^{2} E = μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}}$ . The wave speed is $v = 1/ μ_{0} ϵ_{0} = c$ . The same argument applied to the curl of the Ampere-Maxwell law gives the same equation for $B$ . $□$

Connections Master

Faraday's law 10.03.02 is the other dynamical equation; the Ampere-Maxwell law and Faraday's law form the dynamical pair.
EM waves 10.04.02 are the direct consequence of the coupled $E$ and $B$ equations.
Poynting vector 10.03.05 describes energy flow; Poynting's theorem is derived from the complete Maxwell equations.
Plane waves in matter 10.04.03 use the Maxwell equations in material media.
Faraday tensor 10.06.01 is the covariant formulation; $d F = 0$ and $d * F = J$ encode all four equations.
Larmor formula 10.07.01 for radiation from an accelerating charge follows from the retarded solutions of the Maxwell equations.

Historical & philosophical context Master

Maxwell introduced the displacement current in his 1861 paper "On Physical Lines of Force." His original motivation was mechanical: he modelled the electromagnetic field as an elastic medium (the "luminiferous aether") and the displacement current arose naturally from the elastic properties of this medium. The aether interpretation was later abandoned, but the displacement current remained.

The key prediction was that electromagnetic waves travel at speed $c = 1/ μ_{0} ϵ_{0}$ , which Maxwell calculated from the known values of $μ_{0}$ and $ϵ_{0}$ and found to equal the speed of light. This was the first unification of optics with electromagnetism: light is an electromagnetic wave.

The continuity equation (charge conservation) was known before Maxwell, but its connection to the displacement current was Maxwell's insight. In modern physics, charge conservation follows from gauge invariance (Noether's theorem), and the displacement current is the manifestation of this conservation law in the field equations.

Bibliography Master

Maxwell, J. C., "On Physical Lines of Force," Phil. Mag. 21, 161-175, 281-291, 338-348 (1861).
Maxwell, J. C., "A Dynamical Theory of the Electromagnetic Field," Phil. Trans. R. Soc. 155, 459-512 (1865).
Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
Sommerfeld, A., Electrodynamics (Academic Press, 1952).

Prerequisites

10.03.01
10.03.02
10.01.04

Used in

10.03.05
10.04.03

Tier anchors

beginner: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 7.3-7.4
intermediate: Griffiths, Introduction to Electrodynamics, 4th ed. (2017), Ch. 7; Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6.2-6.3
master: Jackson, Classical Electrodynamics, 3rd ed. (1999), Ch. 6; Sommerfeld, Electrodynamics (1952), Lectures 10-12

References

Griffiths — Introduction to Electrodynamics, 4th ed. (Cambridge, 2017) · Ch. 7.3 Maxwell's correction to Ampere's law; Ch. 7.4 Maxwell's equations; Ch. 4-7 Boundary conditions in matter
Jackson — Classical Electrodynamics, 3rd ed. (Wiley, 1999) · Ch. 6.2 Displacement current; Ch. 6.3 Maxwell's equations in material media
Sommerfeld — Electrodynamics (Academic Press, 1952) · Lectures 10-12 Maxwell's equations; the displacement current; conservation of charge; energy conservation

Estimated time

beginner: 15m
intermediate: 35m
master: 55m