Ampere-Maxwell Law

Claimed by Rohit Naras (Spring 2026)

The Ampere-Maxwell Law is one of the four Maxwell's Equations and represents a generalization of Ampere's Law. It states that magnetic fields are produced not only by electric currents but also by time-varying electric fields (the so-called "displacement current"). This addition by James Clerk Maxwell was the missing piece that unified electricity and magnetism, and it directly predicts the existence of electromagnetic waves traveling at the speed of light.

The Main Idea

Ampere's original law relates the magnetic field along a closed (Amperian) loop to the conduction current piercing any surface bounded by that loop. This works perfectly for steady currents in continuous wires, but it fails in a famous test case: a charging capacitor. If you draw an Amperian loop around the wire feeding the capacitor and choose a flat surface, you enclose a current I. But if you bulge the surface so it passes between the capacitor plates instead, no conduction current pierces it at all — yet the magnetic field around the loop is the same. Ampere's Law gives two different answers for the same loop, which is a contradiction.

Maxwell resolved this by recognizing that between the plates, the electric field is changing in time, and this changing E-field must itself act as a source of magnetic field. He added a new term — the displacement current — proportional to the rate of change of electric flux. The corrected law is symmetric with Faraday's Law: just as a changing magnetic flux produces a curly electric field, a changing electric flux produces a curly magnetic field.

This symmetry is not just aesthetic. When you combine the Ampere-Maxwell Law with Faraday's Law, the equations support self-sustaining oscillations of E and B fields propagating through space at speed [math]\displaystyle{ c = 1/\sqrt{\mu_0 \epsilon_0} \approx 3 \times 10^8 }[/math] m/s. Light is an electromagnetic wave, and the Ampere-Maxwell Law is half the reason why.

A Mathematical Model

The full Ampere-Maxwell Law is:

[math]\displaystyle{ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} }[/math]

where:

• [math]\displaystyle{ \oint \vec{B} \cdot d\vec{l} }[/math] is the line integral of the magnetic field around a closed Amperian loop
• [math]\displaystyle{ I_{enc} }[/math] is the conduction current passing through any surface bounded by that loop
• [math]\displaystyle{ \Phi_E = \int \vec{E} \cdot \hat{n}\, dA }[/math] is the electric flux through that same surface
• [math]\displaystyle{ \mu_0 = 4\pi \times 10^{-7} }[/math] T·m/A is the permeability of free space
• [math]\displaystyle{ \epsilon_0 = 8.854 \times 10^{-12} }[/math] C²/(N·m²) is the permittivity of free space

The second term, [math]\displaystyle{ \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} }[/math], is Maxwell's contribution. The quantity [math]\displaystyle{ I_d = \epsilon_0 \frac{d\Phi_E}{dt} }[/math] is called the displacement current, though it is not a true current of moving charges, it is a name given to a changing electric flux that has the same magnetic effect as a real current.

Computational Model

The simulation below uses VPython/GlowScript to visualize the Ampere-Maxwell Law for a charging parallel-plate capacitor. As the capacitor charges, the electric field between the plates grows, producing a changing electric flux. By the Ampere-Maxwell Law, this changing flux generates a curly magnetic field between the plates — even though no real current flows there.

The simulation shows:

• Two circular capacitor plates being charged by a current I
• A growing electric field [math]\displaystyle{ \vec{E} }[/math] (red arrow) between the plates
• The induced curly magnetic field [math]\displaystyle{ \vec{B} }[/math] (blue arrows) circulating around the field axis
• Real-time graphs of [math]\displaystyle{ E(t) }[/math], [math]\displaystyle{ dE/dt }[/math], and the resulting [math]\displaystyle{ B(r) }[/math] at a fixed radius

You can run the simulation here: [Insert your GlowScript link after publishing — instructions below]

The full GlowScript code is also provided below for transparency and reproducibility.

Link: https://trinket.io/glowscript/faf9ad543e9a

Examples

Simple

Question: A 30 A current is charging a parallel-plate capacitor with circular plates of radius R = 2 mm. The plate separation is 5 mm. What is the magnitude of the magnetic field between the plates at a radial distance r = 1 mm from the central axis?

Solution:

Apply the Ampere-Maxwell Law to a circular Amperian loop of radius r = 1 mm centered on the axis between the plates. There is no conduction current piercing this loop (we are between the plates, not in the wire), so only the displacement current term contributes:

[math]\displaystyle{ \oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} }[/math]

The electric field between the plates of a parallel-plate capacitor is uniform across the plate area:

[math]\displaystyle{ E = \frac{Q}{\epsilon_0 A_{plate}} = \frac{Q}{\epsilon_0 \pi R^2} }[/math]

The electric flux through our small Amperian loop of area [math]\displaystyle{ \pi r^2 }[/math] is:

[math]\displaystyle{ \Phi_E = E \cdot \pi r^2 = \frac{Q}{\epsilon_0 \pi R^2} \cdot \pi r^2 = \frac{Q r^2}{\epsilon_0 R^2} }[/math]

Taking the time derivative and recognizing [math]\displaystyle{ dQ/dt = I }[/math]:

[math]\displaystyle{ \frac{d\Phi_E}{dt} = \frac{I r^2}{\epsilon_0 R^2} }[/math]

Substituting back into the Ampere-Maxwell Law:

[math]\displaystyle{ \oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \cdot \frac{I r^2}{\epsilon_0 R^2} = \frac{\mu_0 I r^2}{R^2} }[/math]

By the symmetry of the problem, B is constant in magnitude along the Amperian circle and is everywhere tangent to it, so the line integral simplifies to [math]\displaystyle{ B(2\pi r) }[/math]:

[math]\displaystyle{ B(2\pi r) = \frac{\mu_0 I r^2}{R^2} }[/math]

Solving for B:

[math]\displaystyle{ B = \frac{\mu_0 I r}{2\pi R^2} }[/math]

Plugging in I = 30 A, r = 0.001 m, R = 0.002 m:

[math]\displaystyle{ B = \frac{(4\pi \times 10^{-7}\ \text{T·m/A})(30\ \text{A})(0.001\ \text{m})}{2\pi (0.002\ \text{m})^2} }[/math]

[math]\displaystyle{ B = \frac{(4\pi \times 10^{-7})(0.03)}{2\pi (4 \times 10^{-6})} = 1.5 \times 10^{-3}\ \text{T} }[/math]

[math]\displaystyle{ \boxed{B = 1.5\ \text{mT}} }[/math]

Middling

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.

At a time change in t, we can calculate the area as the height (h) times the speed over the change in time. ⩟ A = v (⩟ t) h Because the electric field is constant in this region we can also calculate the change in electric flux over time as Evh(⩟ t) /(⩟ t) which is the same as Evh Calculating the path integral for the magnetic field we get that ∮B . dl = Bh cos 0 = Bh An important thing to notice is that there is no current I, so, using the Ampere-Maxwell Law we can see that

Bh = μ. [I+ ε. (vEh)]

but since there is no current,

B = μ. ε. (vE)

From Faraday's law we get that the emf equals the rate of change of the magnetic flux: Eh = Bvh

Substituting E = Bv into our previous equation we get that

B = μ. ε. (v(vB))

Solving for v we get that:

The example shows that the speed of light relates the a time varying electric and magnetic field.

What this implies

In the example above we saw that the speed of light relates a time varying electric and magnetic field. This translates to the fact that an electromagnetic wave propagates at the speed of light.

Difficult

Question 1: First find the line integral of B around a loop of radius R located just outside the left capacitor plate.

Answer:

[math]\displaystyle{ \int \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 I(t) }[/math]

Question 2: Now find an expression for the same line integral of B around the same loop located just outside the left capacitor plate as before. Use the surface that passes between the plates of the capacitor, where there is no conduction current. This should be found by evaluating the amount of displacement current in the Ampère-Maxwell law above.

Answer:

[math]\displaystyle{ \Phi_E = \int E \cdot dA = A E(t) }[/math]

[math]\displaystyle{ I = ε_0 d \frac{(AE(t))}{dt} = A ε_0 \frac{dE(t)}{dt} }[/math]

[math]\displaystyle{ \int \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 I }[/math]

[math]\displaystyle{ \int \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 A ε_0 \frac{dE(t)}{dt} }[/math]

Question 3: Express the normal current in terms of the charge on the capacitor plate.

Answer:

[math]\displaystyle{ I = ε_0 \frac{d\frac{q}{ε_0}}{dt} }[/math]

the expression for the normal current is:

[math]\displaystyle{ I = \frac{dq(t)}{dt} }[/math]

the current is directly related to charge on left capacitor plate

Conceptual Questions

The plot above shows a side and a top view of a capacitor with charge Q with electric and magnetic fields E and B at time t. The charge Q is:

1) Increasing in time

2) Decreasing in time

3) Constant in time

4) I don't know

Answer:

The unit vector N points out of the plane. Given that the magnetic field curls clockwise, the electric flux would be positive and decreasing. Hence E is decreasing. Thus Q must be decreasing, since E is proportional to Q.

Consider a circular capacitor, with an Amperian circular loop (radius r) in the plane midway between the plates. When the capacitor is charging, the line integral of the magnetic field around the circle (in direction shown) is

1) Zero

2) Positive

3) Negative

4) Can’t tell

Answer:

One thing to notice is that there is no enclosed current through the disk. When integrating in the direction shown, the electric flux is positive. Because the plates are charging, the electric flux is increasing. Therefore the line integral is positive.

Connectedness

How is this topic connected to something that you are interested in?

As a computer engineering student, the Ampere-Maxwell Law is foundational to almost every piece of hardware I work with. Every PCB trace carrying a high-frequency signal radiates because of changing electric fields between adjacent conductors, this is exactly the displacement current term in action. Signal integrity engineering, antenna design, and even the GHz-range coupling that causes crosstalk in dense chip layouts are all governed by this equation. Without Maxwell's correction, we would have no theoretical basis for wireless communication, RF circuits, or the high-speed serial links inside modern processors.

Interesting industrial application

A particularly elegant application is the wireless charging standard used in modern smartphones (Qi). Two flat coils: one in the charging pad, one in the phone, couple through a time-varying magnetic field. But at the higher-frequency variants (resonant inductive and especially capacitive wireless power transfer), the displacement current term becomes the dominant coupling mechanism. Capacitive wireless power transfer literally relies on the Ampere-Maxwell Law: there is no closed conductor between transmitter and receiver, only a changing electric flux that carries the energy across the gap.

History

Andre-Marie Ampere formulated his original law in 1826, relating magnetic fields to steady electric currents. For the next ~35 years it was treated as complete. The problem was that Ampere's Law, in its original form, only worked for closed steady currents — and a charging capacitor is the canonical case where it visibly fails.

In the early 1860s, James Clerk Maxwell was working to unify the existing laws of electricity and magnetism into a single mathematical framework. While studying Faraday's work on induction, Maxwell noticed the asymmetry: a changing magnetic field produced an electric field (Faraday's Law), but no equivalent law existed for a changing electric field producing a magnetic field. In his 1861 paper On Physical Lines of Force and more rigorously in his 1865 paper A Dynamical Theory of the Electromagnetic Field, Maxwell introduced the displacement current term to fix Ampere's Law and restore symmetry.

The payoff was immediate and stunning. With the corrected equation, Maxwell could derive a wave equation directly from his four laws, and the predicted wave speed [math]\displaystyle{ 1/\sqrt{\mu_0\epsilon_0} }[/math] came out to within experimental error of the measured speed of light. Maxwell concluded — correctly — that light itself is an electromagnetic wave. Heinrich Hertz experimentally confirmed the existence of these waves in 1887, and the entire modern world of radio, television, radar, and wireless communication followed.

See also

The Ampere-Maxwell Law relates to the other Maxwell equations:

• Gauss's Law

• Magnetic Flux

• Faraday's Law

It might also help to read up on Maxwell's Electromagnetic Theory

Also for continuity purposes, it can be helpful to look at the incomplete Ampere's Law:

• Ampere's Law

Further Reading

Matter and Interactions: Volume 2 by Ruth Chabay and Bruce Sherwood (4th Edition)

External Links

Maxwell's Equations: http://study.com/academy/lesson/maxwells-equations-definition-application.html

James Maxwell Biography: http://www.biography.com/people/james-c-maxwell-9403463

References

http://bulldog2.redlands.edu/fac/eric_hill/Phys232/Lectures/Ch%2023%20lect%201.pdf http://ocw.mit.edu/high-school/physics/exam-prep/electromagnetism/maxwells-equations/ http://ocw.mit.edu/courses/physics/8-02-physics-ii-electricity-and-magnetism-spring-2007/readings/summary_w13d1.pdf http://www.schoolphysics.co.uk/age16-19/Wave%20properties/Wave%20properties/text/Electromagnetic_radiation/index.html