Ampère's circuital law

Template:Redirect Template:Electromagnetism In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law that André-Marie Ampère discovered in 1823) relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell (not Ampère) derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

Maxwell's original circuital law

The original form of Maxwell's circuital law, which he derived in his 1855 paper "On Faraday's Lines of Force" based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them. It determines the magnetic field associated with a given current, or the current associated with a given magnetic field.

The original circuital law is only a correct law of physics in a magnetostatic situation, where the system is static except possibly for continuous steady currents within closed loops. 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).

Equivalent forms

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 [[magnetic field|Template:Math or Template:Math magnetic fields]]. These two forms use the total current density and free current density, respectively. The Template:Math and Template:Math fields are related by the constitutive equation: Template:Math where Template:Math is the magnetic constant.

Explanation

The integral form of the original circuital law is a line integral of the magnetic field around some closed curve Template:Mvar (arbitrary but must be closed). The curve Template:Mvar in turn bounds both a surface Template:Mvar which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by Template:Mvar), 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).

In terms of total current, (which is the sum of both free current and bound current) the line integral of the [[magnetic field|magnetic Template:Math-field]] (in teslas, T) around closed curve Template:Mvar is proportional to the total current Template:Math passing through a surface Template:Mvar (enclosed by Template:Mvar). In terms of free current, the line integral of the [[magnetic field|magnetic Template:Math-field]] (in amperes per metre, A·m⁻¹) around closed curve Template:Mvar equals the free current Template:Math through a surface Template:Mvar.

Forms of the original circuital law written in SI units

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

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

Ambiguities and sign conventions

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 Template:Math could go around the loop in either direction (clockwise or counterclockwise); the vector area Template:Math could point in either of the two directions normal to the surface; and Template:Math is the net current passing through the surface Template:Mvar, 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 Template:Math. Also the current passing in the same direction as Template:Math 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 Template:Mvar that have the curve Template:Mvar 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 Template:Mvar, 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.

Free current versus bound current

The electric current that arises in the simplest textbook situations would be classified as "free current"—for example, the current that passes through a wire or battery. In contrast, "bound current" arises in the context of bulk materials that can be magnetized and/or polarized. (All materials can to some extent.)

When a material is magnetized (for example, by placing it in an external magnetic field), the electrons remain bound to their respective atoms, but behave as if they were orbiting the nucleus in a particular direction, creating a microscopic current. When the currents from all these atoms are put together, they create the same effect as a macroscopic current, circulating perpetually around the magnetized object. This magnetization current Template:Math is one contribution to "bound current".

The other source of bound current is bound charge. When an electric field is applied, the positive and negative bound charges can separate over atomic distances in polarizable materials, and when the bound charges move, the polarization changes, creating another contribution to the "bound current", the polarization current Template:Math.

The total current density Template:Math due to free and bound charges is then:

$\mathbf {J} =\mathbf {J} _{\mathrm {f} }+\mathbf {J} _{\mathrm {M} }+\mathbf {J} _{\mathrm {P} }\,,$

with Template:Math  the "free" or "conduction" current density.

All current is fundamentally the same, microscopically. Nevertheless, 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 circuital law, expressed in terms of Template:Math and the microscopic current (which includes free, magnetization and polarization currents), is sometimes put into the equivalent form below in terms of Template:Math 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.

Shortcomings of the original formulation of the circuital law

There are two important issues regarding the circuital law that require closer scrutiny. First, there is an issue regarding the continuity equation for electrical charge. In vector calculus, the identity for the divergence of a curl states that the divergence of the curl of a vector field must always be zero. Hence

$\nabla \cdot (\nabla \times \mathbf {B} )=0\,,$

and so the original Ampère's circuital law implies that

$\nabla \cdot \mathbf {J} =0\,.$

But in general, reality follows the continuity equation for electric charge:

$\nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}\,,$

which is nonzero 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

$\mathbf {J} =\mathbf {0} \,.$

The circuital law implies that

$\nabla \times \mathbf {B} =\mathbf {0} \,,$

but to maintain consistency with the continuity equation for electric charge, we must have

$\nabla \times \mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\,.$

To treat these situations, the contribution of displacement current must be added to the current term in the circuital 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 hydrodynamically and mechanically. He added this displacement current to Ampère's circuital law at equation 112 in his 1861 paper "On Physical Lines of Force".

Displacement current

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 Template:Math. A changing state of polarization is equivalent to a current.

Both contributions to the displacement current are combined by defining the displacement current as:

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

where the electric displacement field is defined as:

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

where Template:Math is the electric constant, Template:Math the relative static permittivity, and Template:Math is the polarization density. Substituting this form for Template:Math in the expression for displacement current, it has two components:

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

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.

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.

Extending the original law: the Maxwell–Ampère equation

Next, the circuital equation is extended by including the polarization current, thereby remedying the limited applicability of the original circuital law.

Treating free charges separately from bound charges, The equation including Maxwell's correction in terms of the Template:Math-field is (the Template:Math-field is used because it includes the magnetization currents, so Template:Math does not appear explicitly, see [[Magnetic Field#Physical interpretation of the H field|Template:Math-field]] and also Note):

$\oint _{C}\mathbf {H} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S}$

(integral form), where Template:Math is the [[magnetic field|magnetic Template:Math field]] (also called "auxiliary magnetic field", "magnetic field intensity", or just "magnetic field"), Template:Math is the electric displacement field, and Template:Math is the enclosed conduction current or free current density. In differential form,

$\mathbf {\nabla } \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\,.$

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 in integral form (see the "proof" section below):

In differential form,

In both forms Template:Math 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:

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

where current density Template:Math is the displacement current, and Template:Math is the current density contribution actually due to movement of charges, both free and bound. Because Template:Math, the charge continuity issue with Ampère's original formulation is no longer a problem. Because of the term in Template:Math, 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.

Proof of equivalence

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

Ampère's circuital law in cgs units

In cgs units, the integral form of the equation, including Maxwell's correction, reads

$\oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}={\frac {1}{c}}\iint _{S}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} \,,$

where Template:Mvar is the speed of light.

The differential form of the equation (again, including Maxwell's correction) is

$\mathbf {\nabla } \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)\,.$

See also

• Biot–Savart law
• Displacement current
• Capacitance
• Ampèrian magnetic dipole model
• Electromagnetic wave equation
• Maxwell's equations

• Faraday's law of induction
• Bound charge
• Electric current
• Vector calculus
• Stokes' theorem

Further reading

• Template:Cite book
• Template:Cite book

External links

• MISN-0-138 Ampere's Law (PDF file) by Kirby Morgan for Project PHYSNET.
• MISN-0-145 The Ampere–Maxwell Equation; Displacement Current (PDF file) by J.S. Kovacs for Project PHYSNET.
• A Dynamical Theory of the Electromagnetic Field Maxwell's paper of 1864