Displacement Current and Maxwell's Equations
Ampère's law, as the static chapters left it, was quietly broken — it failed the simplest test of charge conservation. Maxwell's repair was a single extra term, the displacement current, and with it the four equations of electromagnetism finally closed on themselves. Out of that closure fell a prediction no one had asked for: that disturbed fields ride off together at the speed of light.
- Why the static Ampère law contradicts charge conservation.
- Maxwell's displacement current \(\partial\vec{D}/\partial t\) and the capacitor paradox it resolves.
- The Ampère–Maxwell law \(\nabla\times\vec{H}=\vec{J}+\partial\vec{D}/\partial t\).
- The complete set of four Maxwell equations in point and integral form.
- Boundary conditions for time-varying fields, including the perfect conductor.
- How the equations predict electromagnetic waves at \(c=1/\sqrt{\mu_0\varepsilon_0}\).
The Flaw in Ampère's Law
The static Ampère law of Chapter 12 reads \(\nabla\times\vec{H}=\vec{J}\). Take the divergence of both sides. The left side vanishes identically — the divergence of any curl is zero — which forces \(\nabla\cdot\vec{J}=0\). But charge conservation, the continuity equation, says something different whenever charge is piling up:
The two agree only in statics, where \(\partial\rho_v/\partial t=0\). The cleanest illustration is a charging capacitor: choose an Amperian loop around the lead wire. Cap it with a flat disk and a conduction current \(I\) pierces it; bulge that same surface forward so it passes between the plates and no conduction current crosses it — yet the loop is unchanged. Ampère's law returns two different answers. Something must flow across the gap.
Displacement Current
Maxwell's insight was that the changing electric field in the gap plays the role of the missing current. As charge builds on the plates, \(\vec{D}\) between them rises, and its time rate \(\partial\vec{D}/\partial t\) is a current density in its own right — the displacement current density:
Between the plates \(D=\rho_s=Q/A\), so \(\partial D/\partial t=(1/A)\,dQ/dt=I/A\). Integrating over the plate gives \(I_d=I\) exactly — the displacement current across the gap equals the conduction current in the wire. No charge actually crosses the gap; it is the field that carries the continuity through. This term is tiny in ordinary circuits but dominant at high frequency, and it is the engine of every antenna and waveguide.
The displacement current restores charge conservation and makes magnetism respond to a changing \(\vec{E}\), exactly as Faraday made electricity respond to a changing \(\vec{B}\). It is the missing half of the symmetry.
The Ampère–Maxwell Law
Adding the new term to Ampère's law gives the corrected curl equation. Now take its divergence: the curl term still vanishes, leaving \(\nabla\cdot\vec{J}+\partial(\nabla\cdot\vec{D})/\partial t=\nabla\cdot\vec{J}+\partial\rho_v/\partial t=0\) — continuity is satisfied automatically, for free.
Read it alongside Faraday's \(\nabla\times\vec{E}=-\partial\vec{B}/\partial t\) and the symmetry is unmistakable: a changing \(\vec{B}\) curls up an \(\vec{E}\), and a changing \(\vec{E}\) curls up an \(\vec{H}\). The two fields now feed each other, and that mutual feeding is what lets a disturbance propagate with no charges or wires to carry it.
Maxwell's Equations
With the displacement current in place, the four laws of the whole course stand complete. Each says one thing about the sources of divergence and curl of the fields:
| Law | Point form | Integral form |
|---|---|---|
| Gauss (electric) | \(\nabla\cdot\vec{D}=\rho_v\) | \(\oint\vec{D}\cdot d\vec{S}=Q_{\text{enc}}\) |
| Gauss (magnetic) | \(\nabla\cdot\vec{B}=0\) | \(\oint\vec{B}\cdot d\vec{S}=0\) |
| Faraday | \(\nabla\times\vec{E}=-\dfrac{\partial\vec{B}}{\partial t}\) | \(\oint\vec{E}\cdot d\vec{l}=-\dfrac{d\Phi}{dt}\) |
| Ampère–Maxwell | \(\nabla\times\vec{H}=\vec{J}+\dfrac{\partial\vec{D}}{\partial t}\) | \(\oint\vec{H}\cdot d\vec{l}=I+\dfrac{d\Psi_e}{dt}\) |
They are completed by the constitutive relations that describe the medium — \(\vec{D}=\varepsilon\vec{E}\), \(\vec{B}=\mu\vec{H}\), \(\vec{J}=\sigma\vec{E}\) — from Chapters 7 and 13. Two divergence laws fix the sources of the fields; two curl laws fix how they change in time. Everything in Parts 5 through 7 — waves, transmission lines, waveguides, antennas — is just these four equations applied to a particular geometry.
Boundary Conditions
At an interface between two media the fields must obey matching rules, obtained by shrinking Gaussian pillboxes and Amperian loops onto the surface. For time-varying fields the relations keep the very same form as in statics, because the fields stay finite:
Tangential \(\vec{E}\) and normal \(\vec{B}\) are always continuous; tangential \(\vec{H}\) jumps by any free surface current \(K\), and normal \(\vec{D}\) jumps by any free surface charge \(\rho_s\). At the surface of a perfect conductor the interior fields vanish, so outside it \(E_t=0\) and \(B_n=0\): the field must meet the metal head-on, supported by induced surface charge and current. These conditions are exactly what shape the modes of a waveguide and the reflection of a wave in Parts 6 and 7.
The Prediction of Waves
In a source-free linear medium (\(\vec{J}=0\), \(\rho_v=0\)), take the curl of Faraday's law and substitute Ampère–Maxwell. Using \(\nabla\times(\nabla\times\vec{E})=\nabla(\nabla\cdot\vec{E})-\nabla^2\vec{E}\) with \(\nabla\cdot\vec{E}=0\), the two curl equations fold into a single second-order equation:
Plug in the measured \(\mu_0\) and \(\varepsilon_0\) — two constants from electrostatics and magnetostatics — and out comes the speed of light. Maxwell concluded that light is an electromagnetic wave, the deepest single result in classical physics, and the doorway to Part 5.
Worked Examples
Problem. A parallel-plate capacitor (area \(0.01\ \text{m}^2\), gap \(1\ \text{mm}\), air) has \(v(t)=50\sin(2\pi\cdot5000\,t)\ \text{V}\). Find the peak displacement current.
Solution. \(I_d=C\,dv/dt\) with \(C=\varepsilon_0 A/d\):
Problem. A material has \(\sigma=10^{-4}\ \text{S/m}\), \(\varepsilon_r=5\), at \(f=1\ \text{MHz}\). Find the ratio \(J_c/J_d\) (the loss tangent).
Solution. Use \(J_c/J_d=\sigma/\omega\varepsilon\):
Problem. Compute the wave speed in vacuum from \(\mu_0\) and \(\varepsilon_0\).
Solution. Use \(c=1/\sqrt{\mu_0\varepsilon_0}\):
Problem. A circular capacitor carries a uniform displacement current density \(J_d=0.1\ \text{A/m}^2\). Find \(H\) at radius \(r=2\ \text{cm}\) from the axis.
Solution. Ampère–Maxwell with only \(J_d\): \(H(2\pi r)=J_d(\pi r^2)\), so \(H=\tfrac12 r J_d\):
Problem. A region has \(\vec{D}=(3x\,\hat{a}_x+4y\,\hat{a}_y-2z\,\hat{a}_z)\ \text{nC/m}^2\). Find the volume charge density.
Solution. Apply \(\rho_v=\nabla\cdot\vec{D}\):
Problem. In a dielectric (\(\varepsilon_r=2.5\)), \(\vec{E}=5\sin(2\pi\times10^{8}t)\,\hat{a}_x\ \text{V/m}\). Find the peak \(J_d\).
Solution. Use \(J_d=\varepsilon\,\partial E/\partial t\), peak \(=\varepsilon_0\varepsilon_r E_0\omega\):
Chapter Summary
Static \(\nabla\times\vec{H}=\vec{J}\) forces \(\nabla\cdot\vec{J}=0\), contradicting continuity.
\(\vec{J}_d=\partial\vec{D}/\partial t\); equals the conduction current across a capacitor gap.
\(\nabla\times\vec{H}=\vec{J}+\partial\vec{D}/\partial t\); satisfies continuity automatically.
Two divergence (sources), two curl (time change); plus \(\vec{D}=\varepsilon\vec{E}\), \(\vec{B}=\mu\vec{H}\), \(\vec{J}=\sigma\vec{E}\).
\(E_{1t}=E_{2t}\), \(B_{1n}=B_{2n}\); \(H\) and \(D\) jump by \(K\) and \(\rho_s\); conductor: \(E_t=0\).
\(\nabla^2\vec{E}=\mu\varepsilon\,\partial^2\vec{E}/\partial t^2\); \(c=1/\sqrt{\mu_0\varepsilon_0}\); light is electromagnetic.
Problems
For each item, identify which Maxwell equation or relation applies, then substitute carefully. Difficulty rises down the list.
- A capacitor (area \(20\ \text{cm}^2\), gap \(0.5\ \text{mm}\), air) has \(dv/dt=10^{6}\ \text{V/s}\). Find the displacement current.
- Show, by taking the divergence of the Ampère–Maxwell law, that it implies the continuity equation.
- A field \(\vec{D}=D_0\sin(\omega t)\,\hat{a}_z\) fills a region. Write the displacement current density and its peak value.
- At what frequency are the conduction and displacement currents equal in a medium with \(\sigma=5\times10^{-3}\ \text{S/m}\), \(\varepsilon_r=80\)?
- Given \(\vec{D}=(2x\,\hat{a}_x-3y\,\hat{a}_y+z\,\hat{a}_z)\ \text{nC/m}^2\), find \(\rho_v\).
- Verify that \(\vec{B}=(2y\,\hat{a}_x-3x\,\hat{a}_y)\ \text{T}\) is a legitimate magnetic field by testing \(\nabla\cdot\vec{B}=0\).
- Find the wave speed and refractive index in a non-magnetic dielectric with \(\varepsilon_r=4\).
- A \(1\ \text{GHz}\) field has \(E_0=100\ \text{V/m}\) in free space. Find the peak displacement current density.
- For a circular parallel-plate capacitor of radius \(a\), show that the induced \(H\) at the plate edge is \(H=\tfrac12 a\,\partial D/\partial t\).
- State the boundary conditions on \(\vec{E}\) and \(\vec{H}\) at the surface of a perfect conductor, and explain why \(E_t=0\).
- Starting from the four equations in a source-free medium, derive the wave equation for \(\vec{H}\).
- Explain physically why the displacement current is negligible at \(50\ \text{Hz}\) in a copper wire but dominant in the gap of a microwave capacitor.