Displacement current and waves

Charge a capacitor and something strange happens to Ampere's circuital law. Draw one Amperian loop that cuts the connecting wire, and it sees a current. Draw another loop, bulging through the gap between the plates where no charge ever crosses, and it sees none — yet both loops bound the same circuit. James Clerk Maxwell spotted this contradiction and fixed it with a bold idea: a changing electric field between the plates acts exactly like a current, even though no charge moves. He called it displacement current. The fix did far more than patch a law — it revealed a hidden symmetry between electric and magnetic fields that lets a purely electromagnetic disturbance regenerate itself and travel through empty space at the speed of light. This article builds the corrected law, works two capacitor numericals CISCE actually sets, and pins down why the resulting wave must be transverse.

The core insight

A changing electric flux between capacitor plates behaves as a displacement current, $I_d = \varepsilon_0\,d\Phi_E/dt$ , which restores consistency to Ampere's law and, together with Faraday's law, lets oscillating $E$ and $B$ fields sustain each other as a self-propagating transverse electromagnetic wave.

The core idea

Why the old law breaks. Ampere's circuital law, $\oint \vec B\cdot d\vec l = \mu_0 I$ , was built for steady conduction currents in wires. It has no trouble with a straight current-carrying wire — any surface bounded by the loop gives the same enclosed current. But take a circuit with a charging capacitor. A flat surface cutting through the connecting wire encloses the conduction current $I_c$ . A second surface, bulging like a balloon so that it passes between the plates instead of through the wire, encloses no conduction current at all, since no charge physically crosses the gap. Ampere's law, applied to the same loop, now gives two different answers depending purely on which surface you happen to draw. That inconsistency is the crack Maxwell noticed.

Maxwell's patch. Maxwell proposed that whatever current is "missing" in the gap is supplied by the changing electric flux $\Phi_E$ building up there as charge piles onto the plates. He named this equivalent current the displacement current,

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

where: $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$ is the permittivity of free space, $\Phi_E$ is the electric flux through the chosen surface, and $I_d$ has units of current (amperes) even though it involves no moving charge. Ampere's law is then generalised to the Ampere–Maxwell law,

\oint \vec B\cdot d\vec l = \mu_0 (I_c + I_d)

so that any surface bounding the loop — wire-cutting or plate-bulging — gives the same total enclosed current, conduction plus displacement, and $B$ comes out identical either way.

From patched law to travelling wave. Faraday's law already told us a changing $B$ field induces an $E$ field. The Ampere–Maxwell law now says the reverse is also true: a changing $E$ field induces a $B$ field. These two effects can feed each other indefinitely — a changing $E$ regenerates a $B$ , whose own change regenerates the next $E$ , and so on, with no charges or wires needed at all. The disturbance travels outward as an electromagnetic wave, with $\vec E$ and $\vec B$ oscillating perpendicular to each other and both perpendicular to the direction of travel $\vec c$ — the wave is purely transverse, never longitudinal. Maxwell showed this self-sustaining wave must move at $c \approx 3\times10^8\ \text{m/s}$ , exactly the measured speed of light, identifying light itself as an electromagnetic wave.

Figure

A transverse electromagnetic wave with the electric field oscillating vertically the magnetic field oscillating horizontally and the wave travelling forward along the propagation axis

Derivation ADisplacement current keeps Ampere's law self-consistent

The goal of this derivation is to show, for the standard charging-capacitor setup examiners use, that the displacement current in the gap is always exactly equal in magnitude to the conduction current flowing in the wires — which is precisely the condition needed for the Ampere–Maxwell law to give one unique value of $B$ regardless of which surface bounds the Amperian loop.

Step 1. Set up the two surfaces. Take a single Amperian loop that circles the connecting wire somewhere before the capacitor. Surface $S_1$ is flat and cuts straight through the wire; surface $S_2$ bulges past the wire and instead passes through the gap between the plates, touching neither plate. Both surfaces share the same boundary loop, so the Ampere–Maxwell law must give the same $\oint \vec B\cdot d\vec l$ whichever surface is used to evaluate the enclosed current.

Step 2. Evaluate the current through $S_1$ . Since $S_1$ cuts the wire, the only current it sees is the ordinary conduction current flowing to charge the plates: $I_{c}$ on $S_1$ , and $I_d = 0$ there because there is no changing flux to speak of inside a wire in this simple picture.

Step 3. Evaluate the current through $S_2$ . Since $S_2$ passes through the gap, no charge crosses it, so $I_c = 0$ on $S_2$ . But the electric field between the plates is growing as charge accumulates, so the electric flux $\Phi_E$ through $S_2$ is changing, and $I_d = \varepsilon_0\,d\Phi_E/dt$ is nonzero there.

Step 4. Enforce consistency and use charge conservation. Since one plate of the capacitor gains charge at exactly the rate $I_c = dQ/dt$ delivered by the wire, and since $\Phi_E = Q/\varepsilon_0$ for a parallel-plate capacitor (from Gauss's law), differentiating gives $I_d = \varepsilon_0\,d\Phi_E/dt = dQ/dt$ . So $I_d$ on $S_2$ equals $I_c$ on $S_1$ exactly:

I_d = I_c

Both surfaces now enclose the same total current, $I_c + I_d$ — zero displacement plus full conduction on $S_1$ , zero conduction plus full displacement on $S_2$ — so the Ampere–Maxwell law is consistent, and this equality is the working rule used in every capacitor numerical on this topic.

Worked example

A capacitor of capacitance $C = 4.0\ \mu\text{F}$ is being charged so that the potential difference across it rises uniformly at $\dfrac{dV}{dt} = 6.0\times10^{4}\ \text{V/s}$ . Find the displacement current in the gap between the plates.

Step 1. Relate charge to the given rate. Since $Q = CV$ for a capacitor, differentiating with respect to time gives the conduction current flowing in the wires that charge it: $I_c = \dfrac{dQ}{dt} = C\,\dfrac{dV}{dt}$ .

Step 2. Substitute the numbers. $I_c = (4.0\times10^{-6}\ \text{F})(6.0\times10^{4}\ \text{V/s}) = 0.24\ \text{A}$ .

Step 3. Apply the consistency result just derived. Because the displacement current in the gap must equal the conduction current in the wires, $I_d = I_c = 0.24\ \text{A}$ — even though no charge physically crosses the gap, the changing electric flux there behaves exactly like a current of this size for the purposes of the magnetic field it produces just outside the plates.

The displacement current is $I_d = 0.24\ \text{A}$ . This indirect route — find $I_c$ from $dQ/dt$ or $C\,dV/dt$ , then set $I_d = I_c$ — is how CISCE numericals test the idea, rather than by asking students to compute $\Phi_E$ directly.

Common traps

Treating displacement current as real moving charge crossing the gap — it is not; no charge crosses the gap, and $I_d$ is a mathematical stand-in that keeps Ampere's law self-consistent, arising purely from a changing electric flux.
Forgetting that $I_c$ and $I_d$ are equal only for this simple series charging-capacitor circuit, and that they are physically different things (moving charge versus changing flux) occurring in different parts of the circuit.
Missing that displacement current is what makes electromagnetic waves possible in vacuum at all — without it a changing $E$ field could never regenerate a $B$ field where there are no wires or free charges, and light could not exist as a self-propagating wave.
Assuming $\vec E$ or $\vec B$ points along the direction of travel — both are perpendicular to $\vec c$ and to each other, which is exactly why the wave is called transverse, not longitudinal.

Ask the tutor: want to see displacement current computed directly from the electric flux and plate geometry instead of the shortcut $I_d = I_c$ , or explore how the electromagnetic spectrum follows from this same wave picture?

If no charge actually crosses the gap, why does displacement current produce a real magnetic field?

Because the Ampere–Maxwell law treats a changing electric flux exactly as it treats a real conduction current when computing $B$ from a loop integral. Experimentally, the magnetic field measured just outside a charging capacitor's plates matches what you would get by imagining the conduction current continuing straight through the gap. The field is completely real; only the "current" producing it is a flux-based equivalent, not a flow of charge.

Why do we need displacement current to explain light instead of just using Faraday's law?

Faraday's law alone only tells you a changing $B$ field creates an $E$ field — it explains how a magnetic disturbance could generate an electric one. But to let the wave regenerate itself all the way through vacuum, you also need the reverse step: a changing $E$ field creating a $B$ field. That reverse step is exactly what displacement current supplies. Together, Faraday's law and the Ampere–Maxwell law form a closed loop that lets the wave carry itself forward with no charges or currents anywhere along its path.

Is displacement current only relevant inside capacitors, or does it matter elsewhere too?

The capacitor gap is simply the clearest textbook example because the conduction current visibly "disappears." But wherever an electric field changes with time — including empty space carrying a passing electromagnetic wave — a displacement current term exists in principle. It is this general presence, not just the capacitor case, that lets Maxwell's equations describe electromagnetic waves propagating through vacuum, far from any capacitor plates or wires.

Displacement current and waves

The core idea

Worked example

Common traps

Before this

Where this leads