The Ampère-Maxwell Law is a fundamental equation in classical electromagnetism, completing Maxwell’s equations by introducing the displacement current term. This law relates the magnetic field circulation around a closed loop to the total current enclosed, including both conduction current and the time-varying electric field (displacement current). It ensures consistency with the continuity equation and allows electromagnetic waves to propagate in vacuum.
Ampère’s law was originally formulated based on experiments showing that electric currents produce magnetic fields circulating around them. Larger currents were observed to produce stronger magnetic fields. This result is captured in the equation below where the right-hand side counts the total current passing through the surface , and relates it to the total magnetic field measured along the surfaces bounding curve . This surface is arbitrary as long as its boundary matches , and the integral of current density over yields the net current piercing .
Here:
In simple terms, Ampère's law states that the magnetic field produced at , is proportional to the current passing through the surface bounded by . It can also be written in differential form as
Ampère’s law in its original differential form implies that magnetic fields are generated solely by conduction currents. However, this leads to a problem in situations where the current distribution is not continuous or steady — for example, when charges accumulate or deplete in a region, causing the charge density to vary in time.
Consider a parallel-plate capacitor being charged by a current flowing in the connecting wires:
In the wires, there is a conduction current flowing steadily.
Inside the capacitor gap (between the plates), there is no conduction current because it is an insulator (vacuum or dielectric).
Yet, the electric field in the gap changes as the capacitor plates accumulate charge, producing a time-dependent electric flux.
Magnetic field measurements inside the gap show magnetic fields consistent with the current in the wires, even though in the gap.
If Ampère’s law were strictly true as originally formulated, the magnetic field inside the capacitor gap would have to be zero because there is no conduction current there:
This contradicted experimental evidence showing that magnetic fields exist in the gap during capacitor charging.
Because the charge on the capacitor plates changes over time, the charge density in regions near the plates changes, implying can't be zero. However, the continuity equation requires that
meaning conduction current cannot be divergence-free.
Now taking the divergence of Ampère’s law without the displacement current gives
but since the left side is always zero (divergence of a curl is identically zero),
which contradicts the continuity equation!
This contradiction shows Ampère’s law in its original form is incomplete: it cannot account for the magnetic fields created by time-varying electric fields in regions without conduction current, nor can it ensure charge conservation during non-steady processes like capacitor charging.
Maxwell resolved this problem by introducing the displacement current term,
where:
This term accounts for the changing electric field producing a magnetic field, ensuring the law holds even in regions without conduction current.
Taking the divergence of the Ampère-Maxwell equation in differential form:
Since divergence of a curl is zero, and using Gauss's law ,
Rearranging gives the continuity equation:
thus fixing the continuity crisis of Ampere's equation.
Including the displacement current, Ampère-Maxwell law becomes:
Applying Stokes’ theorem to the differential form gives:
Here:
The Ampère-Maxwell law is a cornerstone of classical electromagnetism, introducing the displacement current to complete Ampère’s law. It resolves inconsistencies with charge conservation and enables the description of electromagnetic wave propagation. This law forms one of the four Maxwell’s equations describing the full behavior of electric and magnetic fields.