James Clerk Maxwell derived it using hydrodynamics in his 1861 published paper "On Physical Lines of Force".
[2] In 1865 he generalized the equation to apply to time-varying currents by adding the displacement current term, resulting in the modern form of the law, sometimes called the Ampère–Maxwell law,[3][4][5] which is one of Maxwell's equations that form the basis of classical electromagnetism.
In 1820 Danish physicist Hans Christian Ørsted discovered that an electric current creates a magnetic field around it, when he noticed that the needle of a compass next to a wire carrying current turned so that the needle was perpendicular to the wire.
[6][7] He investigated and discovered the rules which govern the field around a straight current-carrying wire:[8] This sparked a great deal of research into the relation between electricity and magnetism.
The original form of Maxwell's circuital law, which he derived as early as 1855 in his paper "On Faraday's Lines of Force"[9] based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them.
For systems with electric fields that change over time, the original law (as given in this section) must be modified to include a term known as Maxwell's correction (see below).
The original circuital law can be written in several different forms, which are all ultimately equivalent: The integral form of the original circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed).
In terms of free current, the line integral of the magnetic H-field (in amperes per metre, A·m−1) around closed curve C equals the free current If,enc through a surface S.[clarification needed] There are a number of ambiguities in the above definitions that require clarification and a choice of convention.
In contrast, "bound current" arises in the context of bulk materials that can be magnetized and/or polarized.
For example, in free space, where the circuital law implies that i.e. that the magnetic field is irrotational, but to maintain consistency with the continuity equation for electric charge, we must have To treat these situations, the contribution of displacement current must be added to the current term in the circuital law.
[17] He added this displacement current to Ampère's circuital law at equation 112 in his 1861 paper "On Physical Lines of Force".
[18] In free space, the displacement current is related to the time rate of change of electric field.
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.
[19] 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.
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.
Because ∇ ⋅ D = ρ, the charge continuity issue with Ampère's original formulation is no longer a problem.
With the addition of the displacement current, Maxwell was able to hypothesize (correctly) that light was a form of electromagnetic wave.
Proof that the formulations of the circuital law in terms of free current are equivalent to the formulations involving total current In this proof, we will show that the equation is equivalent to the equation Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the Kelvin–Stokes theorem.
In cgs units, the integral form of the equation, including Maxwell's correction, reads where c is the speed of light.