5.9.4. Ampère-Maxwell Law

Table of Contents

From Ampère's Law to Ampère-Maxwell Law

One of Maxwell's great achievements was correcting Ampère's law so that it worked for changing electric fields as well as for electric currents. The result is the Ampère-Maxwell law, one of the four Maxwell equations.

In magnetism, moving electric charge produces a magnetic field. For steady currents, Ampère's law states that the circulation of the magnetic field around a closed loop is proportional to the current passing through the loop. In integral form,

$$
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}
$$

where $I_{\text{enc}}$ is the current enclosed by the loop.

This works well when currents are constant. But it fails in situations where the electric field changes with time, such as in a charging capacitor.

The Capacitor Problem

Consider a capacitor being charged by a current in a wire. Current flows in the wire toward the capacitor plates, but between the plates there is no conduction current through the gap. If we apply the original Ampère's law to a loop around the wire, we get different answers depending on which surface we choose to span the loop.

If the surface cuts through the wire, it encloses current $I$. If the surface bulges out and passes between the capacitor plates, it encloses no conduction current. That would give two different values for the same magnetic circulation, which is impossible.

Maxwell resolved this by introducing an additional term called the displacement current. It is not an actual flow of charges across the gap in the same sense as conduction current in a wire. Instead, it comes from a changing electric field.

Displacement Current

Maxwell defined the displacement current as

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

where $\Phi_E$ is the electric flux through the surface,

$$
\Phi_E = \int \vec{E} \cdot d\vec{A}
$$

and $\varepsilon_0$ is the permittivity of free space.

The Ampère-Maxwell law becomes

$$
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)
$$

This means that magnetic fields are produced both by conduction current and by changing electric flux.

Important form of the Ampère-Maxwell law:
$$
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)
$$
A changing electric field contributes like a current. This contribution is called the displacement current,
$$
I_d = \varepsilon_0 \frac{d\Phi_E}{dt}
$$

Differential Form

In local form, the Ampère-Maxwell law is written as

$$
\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}
$$

Here, $\vec{J}$ is the current density. The curl of the magnetic field depends on two sources, electric current density and time variation of the electric field.

This differential form is especially useful when studying electromagnetic waves and fields in space.

Differential form of the Ampère-Maxwell law:
$$
\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}
$$
The first term comes from conduction current. The second term comes from a changing electric field.

Physical Meaning

The key new idea is that electric and magnetic fields are linked dynamically. A changing electric field creates a magnetic field, just as a changing magnetic field creates an electric field. This symmetry is essential for the existence of electromagnetic waves.

In empty space, where there may be no charges and no currents, the term $\mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$ can still be nonzero. So even in vacuum, a changing electric field can generate a magnetic field.

Charging Capacitor Example

For a parallel plate capacitor of plate area $A$, if the electric field between the plates is approximately uniform, then the electric flux is

$$
\Phi_E = EA
$$

So the displacement current is

$$
I_d = \varepsilon_0 \frac{d(EA)}{dt}
$$

If $A$ is constant,

$$
I_d = \varepsilon_0 A \frac{dE}{dt}
$$

For an ideal charging capacitor, this displacement current equals the conduction current in the wires,

$$
I_d = I
$$

This makes the Ampère-Maxwell law consistent for any surface chosen around the same loop.

For an ideal charging capacitor,
$$
I_d = I
$$
The displacement current between the plates plays the same role in Ampère-Maxwell law as the conduction current in the wire.

Comparison with the Original Ampère's Law

The difference between the older and corrected versions is small in appearance, but profound in meaning.

Law	Equation	Validity
Ampère's law	$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}}$	Steady currents
Ampère-Maxwell law	$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 \left(I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt}\right)$	Steady and changing fields

The added term allows the law to remain valid in time-dependent situations.

Visualizing the Idea

Around a current-carrying wire, magnetic field lines form closed loops. In a charging capacitor, those loops continue to exist even in the region between the plates, because the changing electric field there acts as a source for the magnetic field.

Rendering scientific sketch…

Charging capacitor and displacement current

Why This Law Matters

The Ampère-Maxwell law is not just a correction to an earlier equation. It is one of the central ideas of electromagnetism. It shows that magnetic fields can arise without conduction current, provided the electric field changes with time.

Together with Faraday's law, it leads directly to the prediction that self-sustaining electric and magnetic disturbances can travel through space. Those disturbances are electromagnetic waves.

Central physical message:
A changing electric field produces a magnetic field.
This is the new ingredient introduced by Maxwell, and it is essential for electromagnetic waves.

Summary

The Ampère-Maxwell law extends Ampère's law by adding the displacement current term. In integral form,

$$
\oint \vec{B} \cdot d\vec{\ell} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)
$$

and in differential form,

$$
\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}
$$

Its purpose is to describe magnetic fields correctly when electric fields change with time. The classic example is a charging capacitor, where the displacement current between the plates preserves consistency. This law is one of the foundations of classical electromagnetism.

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