Maxwell’s equations overview

Every electric motor, radio signal, and beam of light obeys the same four lines of vector calculus. Maxwell's equations are the complete differential-form statement of classical electromagnetism: two equations about how charges create diverging field lines, and two about how changing fields curl around and generate each other in time. James Clerk Maxwell did not invent the first three — Gauss and Faraday had already written them down — but he noticed a missing piece in Ampère's law, a term needed purely to keep the equations mathematically consistent with conservation of charge. That single correction, the displacement current, turned four separate laws about electricity and magnetism into a closed, self-consistent field theory, and it is exactly the piece that lets a changing electric field generate a magnetic field with no wire, no charge, and no current in sight — the seed of every electromagnetic wave.

The core insight

Maxwell's four equations say electric field lines diverge from charge, magnetic field lines never diverge (no monopoles), a changing magnetic field curls an electric field (Faraday), and a changing electric field curls a magnetic field via the displacement-current term Maxwell added to Ampère's law — and this last symmetry is what makes self-sustaining electromagnetic waves possible.

The core idea

Four equations, two jobs. In a region with charge density ρ\rho and current density J⃗\vec J (both zero for pure vacuum radiation), Maxwell's equations in differential form are:

∇⋅E⃗=ρε0(Gauss’s law)\nabla\cdot\vec E=\frac{\rho}{\varepsilon_0}\qquad\text{(Gauss's law)}
∇⋅B⃗=0(Gauss’s law for magnetism)\nabla\cdot\vec B=0\qquad\text{(Gauss's law for magnetism)}
∇×E⃗=−∂B⃗∂t(Faraday’s law)\nabla\times\vec E=-\frac{\partial \vec B}{\partial t}\qquad\text{(Faraday's law)}
∇×B⃗=μ0J⃗+μ0ε0∂E⃗∂t(Ampere-Maxwell law)\nabla\times\vec B=\mu_0\vec J+\mu_0\varepsilon_0\frac{\partial \vec E}{\partial t}\qquad\text{(Ampere-Maxwell law)}

where: E⃗\vec E and B⃗\vec B are the electric and magnetic field vectors, ρ\rho is charge density (C/m^3), J⃗\vec J is current density (A/m^2), ε0=8.85×10−12 C2/N m2\varepsilon_0=8.85\times10^{-12}\ \text{C}^2/\text{N m}^2 is the permittivity of free space, and μ0=4π×10−7 T m/A\mu_0=4\pi\times10^{-7}\ \text{T m/A} is the permeability of free space. The two divergence equations are about sources: field lines spread out from electric charge, but never from magnetic charge, because magnetic monopoles do not exist. The two curl equations are about how the fields generate each other in time.

Faraday's half of the symmetry. A time-varying B⃗\vec B produces a circulating, non-conservative E⃗\vec E — this is electromagnetic induction, the working principle behind every generator and transformer. Nothing about this equation is Maxwell's own contribution; Faraday established it experimentally decades earlier.

Maxwell's own contribution. The ordinary Ampère's law reads ∇×B⃗=μ0J⃗\nabla\times\vec B=\mu_0\vec J, relating circulating B⃗\vec B only to real current. Maxwell added the second term, μ0ε0 ∂E⃗/∂t\mu_0\varepsilon_0\,\partial\vec E/\partial t, called the displacement current density, so that a time-varying E⃗\vec E also generates a circulating B⃗\vec B even where J⃗=0\vec J=0. This single addition completes the symmetry between the two curl equations and is the reason the set is called Maxwell's equations rather than Gauss's and Faraday's equations.

Figure
A charging capacitor with conduction current in the wires and displacement current filling the gap, both threading the same circular Amperian loop and producing the same magnetic field
Displacement current completes the circuit through the capacitor gap conduction current J gap, J = 0 displacement current J_d = eps0 dE/dt Amperian loop encircles the gap Curl B around the loop is identical whether you count real current or displacement current

Common traps

  • Thinking the displacement current ε0 ∂E⃗/∂t\varepsilon_0\,\partial\vec E/\partial t is a flow of real charge. It is not: no charge crosses the capacitor gap, yet it acts as a source of B⃗\vec B exactly like a real current does.
  • Conflating ∇⋅B⃗=0\nabla\cdot\vec B=0 with Faraday's law. "No magnetic monopoles" is a separate, independent statement from "a changing B⃗\vec B induces a circulating E⃗\vec E."
  • Applying the source-free vacuum forms (ρ=0,J⃗=0\rho=0,\vec J=0) everywhere. Inside a capacitor's plates or near a current-carrying wire the full equations with ρ,J⃗\rho,\vec J are still needed; only in the gap or far from sources does the source-free form directly apply.
  • Assuming Maxwell discovered all four laws himself. Gauss's and Faraday's laws predate him; his unique, necessary contribution was the displacement-current term in the fourth equation.
Derivation AWhy Ampere's law needed the displacement current

The goal is to show, using only the vector identity that the divergence of any curl is identically zero together with the continuity equation for charge, that Ampère's original law ∇×B⃗=μ0J⃗\nabla\times\vec B=\mu_0\vec J is mathematically inconsistent unless Maxwell's extra term μ0ε0 ∂E⃗/∂t\mu_0\varepsilon_0\,\partial\vec E/\partial t is added, which is why the correction is not optional decoration but a logical necessity forced on Maxwell once he checked his equations against charge conservation.

Step 1. Take the divergence of both sides of the original Ampère's law, ∇×B⃗=μ0J⃗\nabla\times\vec B=\mu_0\vec J. The left side is ∇⋅(∇×B⃗)\nabla\cdot(\nabla\times\vec B), which is identically zero for any vector field B⃗\vec B, because the divergence of a curl always vanishes. So the equation forces ∇⋅J⃗=0\nabla\cdot\vec J=0 everywhere and at all times.

Step 2. Compare that requirement against the continuity equation, the statement of local charge conservation: ∇⋅J⃗+∂ρ∂t=0\nabla\cdot\vec J+\dfrac{\partial\rho}{\partial t}=0. This says ∇⋅J⃗=0\nabla\cdot\vec J=0 only when charge density ρ\rho is not changing in time. In any situation with accumulating charge, such as a capacitor plate being charged, ∂ρ/∂t≠0\partial\rho/\partial t\neq0, so ∇⋅J⃗≠0\nabla\cdot\vec J\neq0, and the original Ampère's law is flatly contradicted.

Step 3. Fix the law by adding a term μ0ε0 ∂E⃗/∂t\mu_0\varepsilon_0\,\partial\vec E/\partial t to the right side: ∇×B⃗=μ0J⃗+μ0ε0 ∂E⃗/∂t\nabla\times\vec B=\mu_0\vec J+\mu_0\varepsilon_0\,\partial\vec E/\partial t. Take the divergence of this corrected equation. The left side is still zero, and using Gauss's law ∇⋅E⃗=ρ/ε0\nabla\cdot\vec E=\rho/\varepsilon_0 on the new term gives 0=μ0∇⋅J⃗+μ0∂ρ∂t0=\mu_0\nabla\cdot\vec J+\mu_0\dfrac{\partial\rho}{\partial t}, which is exactly the continuity equation, now automatically satisfied for any ρ\rho and J⃗\vec J.

Step 4. Read the physical meaning. The extra term, the displacement current density Jd=ε0 ∂E/∂tJ_d=\varepsilon_0\,\partial E/\partial t, behaves as a source of B⃗\vec B even where no real charge moves, such as the gap of a charging capacitor. It "completes the circuit" so that the magnetic field circulating around a wire feeding a capacitor is continuous and consistent, whether the Amperian loop is drawn through the wire or through the gap between the plates.

Worked example

Problem. A parallel-plate capacitor is being charged so the electric field between the plates rises as E(t)=αtE(t)=\alpha t, with α=2×1011 V/(m s)\alpha=2\times10^{11}\ \text{V/(m s)}, and there is no conduction current in the gap (J⃗=0\vec J=0 there). Find (a) the displacement current density JdJ_d in the gap at t=1 μst=1\ \mu\text{s}, and (b) confirm the result is independent of tt.

Step 1. Write the displacement current density from the second term of the Ampere-Maxwell law: Jd=ε0 ∂E/∂t=ε0 αJ_d=\varepsilon_0\,\partial E/\partial t=\varepsilon_0\,\alpha, since E(t)=αtE(t)=\alpha t gives ∂E/∂t=α\partial E/\partial t=\alpha, a constant.

Step 2. Substitute numbers: Jd=(8.85×10−12 C2/N m2)(2×1011 V/(m s))=1.77 A/m2J_d=(8.85\times10^{-12}\ \text{C}^2/\text{N m}^2)(2\times10^{11}\ \text{V/(m s)})=1.77\ \text{A/m}^2.

Step 3. Because α\alpha is constant, ∂E/∂t\partial E/\partial t does not depend on tt, so Jd=1.77 A/m2J_d=1.77\ \text{A/m}^2 at t=1 μst=1\ \mu\text{s} and at every other instant during the charging. The displacement current density is Jd=1.77 A/m2J_d=1.77\ \text{A/m}^2, constant in time.

Competitive stretch

Problem. The same capacitor has circular plates of radius R=5 cmR=5\ \text{cm}. Using the result above, find the magnetic field BB at a point on the edge of the plates (r=Rr=R), by applying the Ampere-Maxwell law to a circular Amperian loop of radius RR centred on the axis, in the gap.

Step 1. The total displacement current through the loop is Id=Jd⋅πR2=(1.77 A/m2)×π×(0.05 m)2=1.39×10−2 AI_d=J_d\cdot\pi R^2=(1.77\ \text{A/m}^2)\times\pi\times(0.05\ \text{m})^2=1.39\times10^{-2}\ \text{A}.

Step 2. By symmetry B⃗\vec B circulates around the axis with constant magnitude on a loop of radius RR, so the Ampere-Maxwell law ∮B⃗⋅dl⃗=μ0Id\oint\vec B\cdot d\vec l=\mu_0 I_d gives B (2πR)=μ0IdB\,(2\pi R)=\mu_0 I_d.

Step 3. Solve for BB: B=μ0Id2πR=(4π×10−7)(1.39×10−2)2π(0.05)=5.56×10−8 TB=\dfrac{\mu_0 I_d}{2\pi R}=\dfrac{(4\pi\times10^{-7})(1.39\times10^{-2})}{2\pi(0.05)}=5.56\times10^{-8}\ \text{T}. The magnetic field at the edge of the plates is about 5.6×10−8 T5.6\times10^{-8}\ \text{T}, generated entirely by the displacement current, with no real current anywhere in the gap.

Ask the tutor: if you want to see the divergence and curl operators worked through on a specific field configuration, or want the wave-equation derivation that follows directly from these four equations, ask the AI tutor about this topic.

Why is the displacement current called a current if no charge actually moves?

It is named a current because it plays exactly the same mathematical role as a real current on the right side of the Ampere-Maxwell law: it produces a circulating magnetic field. Physically, though, it is not charge in motion; it is a changing electric field, ε0 ∂E⃗/∂t\varepsilon_0\,\partial\vec E/\partial t, that acts as a source of B⃗\vec B. The name is a convenience so that the combined quantity J⃗+ε0 ∂E⃗/∂t\vec J+\varepsilon_0\,\partial\vec E/\partial t can be treated as one continuous "total current" that never breaks, even across a capacitor gap where no real charge flows.

If Gauss's and Faraday's laws were already known, why do we call all four Maxwell's equations?

Convention and Maxwell's unifying role, not sole authorship of every line. Gauss's law and Gauss's law for magnetism came from Coulomb's law and experimental observation, and Faraday's law came from Faraday's induction experiments. Maxwell's achievement was twofold: he added the missing displacement-current term that makes the fourth equation consistent with charge conservation, and he assembled all four into one coherent set from which he predicted electromagnetic waves travelling at the speed of light, unifying electricity, magnetism, and optics. The set carries his name because he completed and unified it, not because he wrote every equation from scratch.

Do I need to worry about D and H fields, or media other than vacuum, for this course?

No. This course's scope is vacuum electromagnetism only, with ρ\rho and J⃗\vec J as the only sources, working directly with E⃗\vec E and B⃗\vec B. The auxiliary fields D⃗\vec D and H⃗\vec H used for equations inside dielectric or magnetic materials, along with radiation formalism such as gauges or the Liénard-Wiechert potentials, are explicitly out of scope for PH11001's electromagnetic-waves block and are not tested here.

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