Ampère’s circuital law

Ampère’s circuital law
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Ampère’s_circuital_law".

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Electromagnetism

Electricity · Magnetism

Electrostatics
· Electric charge · Coulomb’s law · Electric field · Electric flux · Gauss’ law · Electric potential · Electrostatic induction · Electric dipole moment ·

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· Ampère’s law · Electric current · Magnetic field · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss’s law for magnetism ·

Electrodynamics
· Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·

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"Ampère's law" redirects here. For the law describing forces between current-carrying wires, see Ampère's force law.

In classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is the magnetic analogue of Gauss's law, and one of the four Maxwell's equations that form the basis of classical electromagnetism.

1 Original Ampère's circuital law
- 1.1 Integral form
- 1.2 Differential form
2 Corrected Ampère's circuital law: the Ampère-Maxwell equation
- 2.1 Displacement current
3 Ampère's law in terms of free current
4 Ampère's law in cgs units
5 See also
6 Further reading
7 External links

Original Ampère's circuital law

An electric current produces a magnetic field.

In its historically original form, Ampère's Circuital law relates the magnetic field $\mathbf{B}$ to its source, the current density $\mathbf{J}$ . The equation is not in general correct (see "Maxwell's correction" below), but is correct in the special case where the electric field is constant (unchanging) in time.

The law can be written in two forms, the "integral form" and the "differential form". The forms are equivalent, and related by the Kelvin-Stokes theorem.

Integral form

In SI units, (the version in cgs units is in a later section), the "integral form" of the original Ampère's Circuital law is:

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S}$

or equivalently,

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_{\mathrm{enc}}$

where:

$\oint_C$ is the closed line integral around contour (closed curve) C.
$\mathbf{B}$ is the magnetic field in teslas.
$\cdot$ is the vector dot product.
$\mathrm{d}\mathbf{l}$ is an infinitesimal element (differential) of the contour C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction equal to the direction of the contour C, see below),
$\iint_S$ denotes an integral over the surface S enclosed by contour C (see below). The double integral sign is meant simply to denote that the integral is two-dimensional in nature.
$\mu_0 \!\$ is the magnetic constant.
$\mathbf{J}$ is the current density, both bound and free, through the surface S enclosed by contour C
$\mathrm{d}\mathbf{S} \!\$ is the vector area of an infinitesimal element of surface S (i.e. 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 discussion),
$I_{\mathrm{enc}} \!\$ is the net current that penetrates through the surface S, both bound and free.

There are a number of ambiguities in the above definitions that warrant elaboration.

First, three of these terms are associated with sign ambiguities: the line integral $\oint_C$ could go around the loop in either direction (clockwise or counterclockwise); the vector area $\mathrm{d}\mathbf{S}$ 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: When the index-finger of the right-hand points along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area $\mathrm{d}\mathbf{S}$ , and current passing in that same direction must be counted as positive. The right hand grip rule can also be used to determine the signs.

Second, there are infinitely many possible surfaces S that have the contour C as their border. (Imagine a soap film on a wire loop, which can be deformed by blowing gently at it.) 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; it can be proven that any surface with boundary C can be chosen.

Differential form

By the Kelvin-Stokes theorem, this equation can also be written in a "differential form". Again, this equation only applies in the case where the electric field is constant in time; see below for the more general form. In SI units, the equation states:

$\mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J}$

where

$\mathbf{\nabla} \times \!\$ is the curl operator.

Corrected Ampère's circuital law: the Ampère-Maxwell equation

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.

The generalized law (in SI units), as corrected by Maxwell, takes the following integral form:

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} + \epsilon_0 \mu_0 {\mathrm{d} \over \mathrm{d}t} \iint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}$

where $\ \epsilon_0$ is the vacuum permittivity and E is the electric field.

This Ampère-Maxwell law can also be stated in differential form (with the Kelvin-Stokes theorem):

$\mathbf{\nabla} \times \mathbf{B} = \mu_0 \mathbf{J} + \epsilon_0 \mu_0 \frac{\partial \mathbf{E}}{\partial t}$

Displacement current

Main article: Displacement current

The displacement current is defined so as to make these equations more transparent. It is defined by

$\mathbf{J}_D=\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

and then the equation is:

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S (\mathbf{J}+\mathbf{J}_D) \cdot \mathrm{d} \mathbf{A}$

(integral form), or

$\mathbf{\nabla}\times \mathbf{B} = \mu_0(\mathbf{J}+\mathbf{J}_D)$

(differential form).

With the addition of the displacement current, Maxwell was able to postulate (correctly) that light was a form of electromagnetic wave. See electromagnetic wave equation for a discussion on this important discovery.

Ampère's law in terms of free current

Note on free current versus bound current

Main article: 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 principally in the context of bulk materials that can be magnetized. (All materials can to some extent.) When such a material is magnetized (e.g., 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 is one source of "bound current". The other source is that there is an analogous thing called bound charge which arises in polarizable materials, and when the polarization changes, the bound charges move, creating another contribution to the "bound current".

In many respects, all current is fundamentally the same, microscopically. Nevertheless, there are often practical reasons for wanting to treat bound current differently from free current. The result is that the more "fundamental" Ampère's law, in terms of B, is sometimes put into the equivalent form below, which is 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.

Integral form

This formulation of Ampère's law states (in SI, including Maxwell's correction):

$\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,enc} + \frac {d}{dt}\iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S}$

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
I_f,enc is the enclosed free current.

Differential form

This formulation of Ampère's law states (in SI, including Maxwell's correction):

$\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$

where J_f is the free current density.

Proof of equivalence

Proof that the formulations of Ampère's law in terms of free current are equivalent to the formulations involving total current.

In this proof, we will show that the equation

$\nabla\times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t}$

is equivalent to the equation

$\mathbf{\nabla} \times \mathbf{B}/\mu_0 = \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Note that we're 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:

$\mathbf{D}=\epsilon_0 \mathbf{E} + \mathbf{P}$

Next, we introduce the magnetization density M, which has the following relation to B and H:

$\mathbf{B}/\mu_0 = \mathbf{H} + \mathbf{M}$

and the following relation to the bound current:

$\mathbf{J}_{\mathrm{bound}} = \nabla\times\mathbf{M} + \frac{\partial \mathbf{P}}{\partial t}$

Now, consider the three equations:

$\mathbf{\nabla} \times \mathbf{M} = \mathbf{J}_{\mathrm{bound}} - \frac{\partial \mathbf{P}}{\partial t}$

$\nabla\times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}}{\partial t}$

$\mathbf{\nabla} \times \mathbf{B}/\mu_0 = \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.

Ampère's law in cgs units

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

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \frac{1}{c} \iint_S \left(4\pi\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}\right) \cdot \mathrm{d}\mathbf{S}$