The Inconsistency in Ampere's Law
An electric current produces a magnetic field around it. To find the field at a point outside a wire, Ampere's circuital law is used in the form
$$\oint \vec{B}\cdot d\vec{l} = \mu_0\, i(t)$$
Maxwell applied this law to a parallel plate capacitor $C$ that is part of a circuit carrying a time-dependent current $i(t)$, and asked for the magnetic field at a point $P$ outside the capacitor. Choosing a plane circular loop of radius $r$ centred symmetrically about the wire, symmetry forces $\vec{B}$ to point along the circumference with the same magnitude everywhere. The line integral becomes $B(2\pi r)$, giving $B(2\pi r) = \mu_0\, i(t)$.
The same loop (red, edge-on) bounds two surfaces. The flat disc $S_1$ is pierced by the wire and encloses the conduction current $i$. The pot-shaped surface $S_2$ dips into the capacitor gap and is pierced by no charge at all.
Now keep the same boundary loop but stretch the surface into a pot shape whose bottom lies between the capacitor plates, or into a tiffin-box shape with the same rim. The left-hand side of the equation, $\oint \vec{B}\cdot d\vec{l}$, is unchanged because the loop is the same. But the right-hand side is now zero, since no conduction current pierces these surfaces. The result is a contradiction: computed one way the field at $P$ is non-zero, computed another way it is zero. Because a magnetic field at a point must have a single value, Ampere's law in its original form is incomplete.
Guessing the Missing Term
The missing term has to restore a single value of $B$ at $P$ regardless of the surface chosen. The clue lies in the tiffin-box surface: although no charge crosses the gap, the electric field does pass through it. For plates of area $A$ holding charge $Q$, the field between them has magnitude $E = (Q/A)/\varepsilon_0$ and is uniform over the area, vanishing outside. The electric flux through that surface, by Gauss's law, is
$$\Phi_E = E A = \frac{Q}{\varepsilon_0}$$
As the capacitor charges, $Q$ changes with time, and the conduction current in the wire is $i = dQ/dt$. Differentiating the flux,
$$\frac{d\Phi_E}{dt} = \frac{1}{\varepsilon_0}\frac{dQ}{dt} \quad\Rightarrow\quad \varepsilon_0\,\frac{d\Phi_E}{dt} = i$$
| Region | Conduction current $i_c$ | Displacement current $i_d$ | Total current |
|---|---|---|---|
| In the connecting wire / outside plates | i | 0 | i |
| In the gap between the plates | 0 | i | i |
This is the term that was missing. Adding $\varepsilon_0$ times the rate of change of electric flux to the conduction current makes the total current identical for every surface bounded by the same loop. With this correction $B$ at $P$ is the same no matter which surface is used, and it equals the field at a point $M$ just inside the gap, exactly as it should.
Defining Displacement Current
The current carried by conductors due to the flow of charges is the conduction current, $i_c$. The new term, due to the changing electric field, is the displacement current, $i_d$:
$$i_d = \varepsilon_0\,\frac{d\Phi_E}{dt}$$
The total current is the sum of the two, $i = i_c + i_d$. NCERT states the regional split explicitly: outside the plates there is only conduction current, $i_c = i$ and $i_d = 0$; inside the gap there is no conduction current, $i_c = 0$, and only displacement current, $i_d = i$. In all respects the displacement current has the same physical effects as the conduction current — in particular, it acts as a source of magnetic field in exactly the same way.
"Displacement" does not mean charges are moving across the gap
The name misleads many students into picturing charge hopping across the capacitor gap. Nothing of the kind happens. There is no flow of charge between the plates. The displacement current is purely the quantity $\varepsilon_0\,\dfrac{d\Phi_E}{dt}$ — it exists because the electric field, and hence the electric flux, is changing in time. It is called a "current" only because it produces the same magnetic effect as a real current.
Remember: $i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$, and for a charging capacitor $|i_d| = |i_c|$ — equal in magnitude, but no charge crosses the gap.
This regeneration of fields is exactly what lets a wave detach and travel. See EM Waves: Source and Nature for how accelerating charges launch them.
The Ampere–Maxwell Law
Generalising Ampere's law to include the displacement current gives the Ampere–Maxwell law, one of the four Maxwell's equations:
$$\oint \vec{B}\cdot d\vec{l} = \mu_0\left(i_c + \varepsilon_0\,\frac{d\Phi_E}{dt}\right)$$
In words: the line integral of the magnetic field around a closed loop equals $\mu_0$ times the total current — conduction plus displacement — threading any surface bounded by that loop. The generalisation makes the source of a magnetic field not just the conduction current of flowing charges, but also the time rate of change of the electric field. The four equations in vacuum are tabulated below.
| Maxwell's equation (vacuum) | Form | Physical statement |
|---|---|---|
| Gauss's law (electricity) | $\oint \vec{E}\cdot d\vec{A} = Q/\varepsilon_0$ | Charge is the source of $\vec{E}$ |
| Gauss's law (magnetism) | $\oint \vec{B}\cdot d\vec{A} = 0$ | No magnetic monopoles |
| Faraday's law | $\oint \vec{E}\cdot d\vec{l} = -\dfrac{d\Phi_B}{dt}$ | Changing $\vec{B}$ gives rise to $\vec{E}$ |
| Ampere–Maxwell law | $\oint \vec{B}\cdot d\vec{l} = \mu_0 i_c + \mu_0\varepsilon_0\dfrac{d\Phi_E}{dt}$ | Current and changing $\vec{E}$ give rise to $\vec{B}$ |
Left: in a wire, charges physically flow — conduction current. Right: in the gap, no charge crosses, but a growing electric field produces a displacement current of equal magnitude. Both are equally valid sources of $\vec{B}$.
Symmetry and Consequences
The displacement current has far-reaching consequences. The first is a new symmetry between electricity and magnetism. Faraday's law, rephrased, says a magnetic field changing with time gives rise to an electric field. The displacement current establishes the mirror image: an electric field changing with time gives rise to a magnetic field. Time-dependent electric and magnetic fields therefore give rise to each other.
A second consequence is that a magnetic field can exist where there is no conduction current at all. In a region with only a time-varying electric field and no flowing charges, the displacement current alone is the source of $\vec{B}$. NCERT notes this can be verified experimentally: the field measured at a point $M$ between the plates of a charging capacitor — where no conduction current exists — equals the field just outside at $P$.
This mutual regeneration is the seed of electromagnetic waves. An oscillating electric field produces an oscillating magnetic field, which in turn produces an oscillating electric field, and the coupled disturbance propagates through space. The full treatment continues in the next section of the chapter.
Where the magnetic field is strongest for a charging capacitor
The displacement-current-filled gap behaves like a uniform "wire" of current. Applying the Ampere–Maxwell law to a circle of radius $r$ inside the plate region, the enclosed displacement current grows as $r^2$, so $B \propto r$ rises linearly outward; beyond the plate edge the full current is enclosed and $B \propto 1/r$ falls. The field therefore peaks at the rim — the imaginary cylindrical surface joining the plate edges — not at the centre and not far outside.
Inside: $B \propto r$. Outside: $B \propto 1/r$. Maximum at the periphery of the plates.
Worked Example
A capacitor of two circular plates, each of radius 12 cm separated by 5.0 cm, is charged by an external source with a constant charging current of 0.15 A. Obtain the displacement current across the plates, and state whether Kirchhoff's junction rule holds at a plate.
The displacement current in the gap equals the conduction current in the wire feeding the plates. For a charging capacitor $i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt} = i_c$, so $$i_d = 0.15\ \text{A}.$$ Kirchhoff's first (junction) rule remains valid at each plate: the conduction current arriving along the wire is exactly continued as displacement current leaving into the gap. Treating the displacement current as the continuation of the conduction current preserves current continuity, so no charge accumulates unaccounted for.
Displacement Current in one screen
- Ampere's law gave two different values of $B$ at one point for a charging capacitor — depending on the surface — so it was incomplete.
- The missing source is the changing electric flux: $i_d = \varepsilon_0\,\dfrac{d\Phi_E}{dt}$.
- For a charging capacitor, $i_d = i_c$ in magnitude; outside the plates $i_d = 0$, inside the gap $i_c = 0$.
- Generalised law: $\oint \vec{B}\cdot d\vec{l} = \mu_0\left(i_c + \varepsilon_0\,\dfrac{d\Phi_E}{dt}\right)$ — the Ampere–Maxwell law.
- Displacement current is not a flow of charge; it is a magnetic-field source arising from a time-varying electric field.
- This completes the symmetry with Faraday's law and predicts electromagnetic waves.