Ampère's circuital law

The original circuital law can be written in several different forms, which are all ultimately equivalent:

• An "integral form" and a "differential form". The forms are exactly equivalent, and related by the Kelvin–Stokes theorem (see the "proof" section below).
• Forms using SI units, and those using cgs units. Other units are possible, but rare. This section will use SI units, with cgs units discussed later.
• Forms using either B or H magnetic fields. These two forms use the total current density and free current density, respectively. The B and H fields are related by the constitutive equation: B = μ₀H in non-magnetic materials where μ₀ is the magnetic constant.

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). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. The mathematical statement of the law is a relation between the total amount of magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral).[10][11]

In terms of total current, (which is the sum of both free current and bound current) the line integral of the magnetic B-field (in teslas, T) around closed curve C is proportional to the total current I_enc passing through a surface S (enclosed by C). 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 I_f,enc through a surface S.

Forms of the original circuital law written in SI units

	Integral form	Differential form
Using B-field and total current	${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\mathrm {enc} }}$	${\displaystyle \mathbf {\nabla } \times \mathbf {B} =\mu _{0}\mathbf {J} }$
Using H-field and free current	${\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\mathbf {J} _{\mathrm {f} }\cdot \mathrm {d} \mathbf {S} =I_{\mathrm {f,enc} }}$	${\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }}$

• J is the total current density (in amperes per square metre, A·m⁻²),
• J_f is the free current density only,
• ∮_C is the closed line integral around the closed curve C,
• ∬_S denotes a 2-D surface integral over S enclosed by C,
• · is the vector dot product,
• dl is an infinitesimal element (a differential) of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C)
• dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule), see below for further explanation of the curve C and surface S.
• ∇ × is the curl operator.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

1. First, three of these terms are associated with sign ambiguities: the line integral ∮_C could go around the loop in either direction (clockwise or counterclockwise); the vector area dS could point in either of the two directions normal to the surface; and I_enc is the net current passing through the surface S, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS. Also the current passing in the same direction as dS must be counted as positive. The right hand grip rule can also be used to determine the signs.
2. Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by moving the wire). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; by Stokes' theorem, the integral is the same for any surface with boundary C, since the integrand is the curl of a smooth field (i.e. exact). In practice, one usually chooses the most convenient surface (with the given boundary) to integrate over.

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 present too, 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:[12]

${\displaystyle \mathbf {J} _{\mathrm {D} }={\frac {\partial }{\partial t}}\mathbf {D} (\mathbf {r} ,\,t)\,,}$

where the electric displacement field is defined as:

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} =\varepsilon _{0}\varepsilon _{\mathrm {r} }\mathbf {E} \,,}$

where ε₀ is the electric constant, ε_r 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:

${\displaystyle \mathbf {J} _{\mathrm {D} }=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\partial \mathbf {P} }{\partial t}}\,.}$

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.[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. 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.

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 ${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}}$ is equivalent to the equation ${\displaystyle {\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,.}$ 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. We introduce the polarization density P, which has the following relation to E and D: ${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \,.}$ Next, we introduce the magnetization density M, which has the following relation to B and H: ${\displaystyle {\frac {1}{\mu _{0}}}\mathbf {B} =\mathbf {H} +\mathbf {M} }$ and the following relation to the bound current: ${\displaystyle {\begin{aligned}\mathbf {J} _{\mathrm {bound} }&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\\&=\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} },\end{aligned}}}$ where ${\displaystyle \mathbf {J} _{\mathrm {M} }=\nabla \times \mathbf {M} ,}$ is called the magnetization current density, and ${\displaystyle \mathbf {J} _{\mathrm {P} }={\frac {\partial \mathbf {P} }{\partial t}},}$ is the polarization current density. Taking the equation for B: ${\displaystyle {\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {\nabla } \times \left(\mathbf {H} +\mathbf {M} \right)\\&=\mathbf {\nabla } \times \mathbf {H} +\mathbf {J} _{\mathrm {M} }\\&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {P} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {J} _{\mathrm {M} }.\end{aligned}}}$ Consequently, referring to the definition of the bound current: ${\displaystyle {\begin{aligned}{\frac {1}{\mu _{0}}}(\mathbf {\nabla } \times \mathbf {B} )&=\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {bound} }+\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\\&=\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}},\end{aligned}}}$ as was to be shown.