All current is fundamentally the same, microscopically. There are often practical reasons for wanting to treat bound current differently from free current. For example, the bound current usually originates over atomic dimensions, and one may wish to take advantage of a simpler theory intended for larger dimensions. The result is that the more microscopic Ampère's law, expressed in terms of B and the microscopic current (which includes free, magnetization and polarization currents), is sometimes put into the equivalent form below in terms of H and the free current only. For a detailed definition of free current and bound current, and the proof that the two formulations are equivalent, see the "proof" section below.
Two issues regarding Ampère's law are important. First, there is an issue regarding the continuity equation for electrical charge. There is a theorem in vector calculus that states the divergence of a curl must always be zero. Hence
and so the original Ampère's law implies that
But in general
which is non-zero for a time-varying charge density. An example occurs in a capacitor circuit where time-varying charge densities exist on the plates.
Second, there is an issue regarding the propagation of electromagnetic waves. For example, in free space, where
Ampère's law implies that
but instead
To treat these situations, the contribution of displacement current must be added to the current term in Ampère's law.
James Clerk Maxwell conceived of displacement current as a polarization current in the dielectric vortex sea, which he used to model the magnetic field with water (hydrodynamically) and with mechanical systems (mechanically). He added this displacement current to Ampère's circuital law at equation (112) in his 1861 paper "" .
In free space, the displacement current is related to the time rate of change of electric field.
In a dielectric the above contribution to displacement current is also present, but a major contribution to the displacement current is related to the polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, the charges in molecules can move a little under the influence of an electric field. The positive and negative charges in molecules separate under the applied field, causing an increase in the state of polarization, expressed as the polarization density P. A changing state of polarization is equivalent to a current.
Both contributions to the displacement current are combined by defining the displacement current as:
where the electric displacement field is defined as:
where ε is the electric constant, "ε" the relative static permittivity, and P is the polarization density. Substituting this form for D in the expression for displacement current, it has two components:
The first term on the right hand side is present everywhere, even in a vacuum. It doesn't involve any actual movement of charge, but it nevertheless has an associated magnetic field, as if it were an actual current. Some authors apply the name "displacement current" to only this contribution.
The second term on the right hand side is the displacement current as originally conceived by Maxwell, associated with the polarization of the individual molecules of the dielectric material.
Maxwell's original explanation for displacement current focused upon the situation that occurs in dielectric media. In the modern post-aether era, the concept has been extended to apply to situations with no material media present, for example, to the vacuum between the plates of a charging vacuum capacitor. The displacement current is justified today because it serves several requirements of an electromagnetic theory: correct prediction of magnetic fields in regions where no free current flows; prediction of wave propagation of electromagnetic fields; and conservation of electric charge in cases where charge density is time-varying. For greater discussion see Displacement current.
Ampère's equation was extended by including the polarization current, thereby remedying the limited applicability of the original Ampère's circuital law.
Treating free charges separately from bound charges, Ampère's equation including Maxwell's correction in terms of the H-field is (the H-field is used because it includes the magnetization currents, so J does not appear explicitly, see H-field and also Note):
(integral form), where H is the magnetic H field (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field", D is the electric displacement field, and J is the enclosed conduction current or free current density. In differential form,
On the other hand, treating all charges on the same footing (disregarding whether they are bound or free charges), the generalized Ampère's equation (also called the Maxwell–Ampère equation) is (see the "proof" section below):
in integral form. In differential form,
In both forms J includes magnetization current density as well as conduction and polarization current densities. That is, the current density on the right side of the Ampère–Maxwell equation is:
where current density J is the "displacement current", and J is the current density contribution actually due to movement of charges, both free and bound. Because , the charge continuity issue with Ampère's original formulation is no longer a problem. Because of the term in "ε∂E / ∂t", wave propagation in free space now is possible.
With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of electromagnetic wave. See electromagnetic wave equation for a discussion of this important discovery.
In cgs units, the integral form of the equation, including Maxwell's correction, reads
where "c" is the speed of light.
The differential form of the equation (again, including Maxwell's correction) is
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