WikiMili The Free Encyclopedia

In classical electromagnetism, **Ampère's circuital law** (not to be confused with Ampère's force law that André-Marie Ampère discovered in 1823)^{ [1] } 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"^{ [2] } and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

- Maxwell's original circuital law
- Equivalent forms
- Explanation
- Ambiguities and sign conventions
- Free current versus bound current
- Shortcomings of the original formulation of the circuital law
- Displacement current
- Extending the original law: the Maxwell–Ampère equation
- Proof of equivalence
- Ampère's circuital law in cgs units
- See also
- Notes
- Further reading
- External links

The original form of Maxwell's circuital law, which he derived in his 1855 paper "On Faraday's Lines of Force"^{ [3] } 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).

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**=*μ*_{0}**H**where*μ*_{0}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).^{ [4] }^{ [5] }

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 currentUsing **H**-field and free current

**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,
- d
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)**l** - d
**S**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.

- 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 d**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: 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 d**S**. Also the current passing in the same direction as d**S**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 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.

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 **J**_{M} 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 **J**_{P}.

The total current density **J** due to free and bound charges is then:

with **J**_{f} 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 **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.

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

and so the original Ampère's circuital law implies that

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

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.^{ [6] }^{ [7] }^{ [8] }^{ [9] }^{ [10] }

Second, there is an issue regarding the propagation of electromagnetic waves. For example, in free space, where

The circuital law implies that

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.

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.^{ [11] } He added this displacement current to Ampère's circuital law at equation 112 in his 1861 paper "On Physical Lines of Force".^{ [12] }

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:^{ [6] }

where the electric displacement field is defined as:

where *ε*_{0} 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:

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.^{ [13] }

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.

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 **H**-field is (the **H**-field is used because it includes the magnetization currents, so **J**_{M} does not appear explicitly, see **H**-field and also Note):^{ [14] }

(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**_{f} 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 in integral form (see the "proof" section below):

In differential form,

In both forms **J** includes magnetization current density^{ [15] } 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**_{D} is the *displacement current*, and **J** is the current density contribution actually due to movement of charges, both free and bound. Because ∇ ⋅ **D** = *ρ*, the charge continuity issue with Ampère's original formulation is no longer a problem.^{ [16] } Because of the term in *ε*_{0}∂**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.

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.

We introduce the polarization density

**P**, which has the following relation to**E**and**D**:Next, we introduce the magnetization density

**M**, which has the following relation to**B**and**H**:and the following relation to the bound current:

where

is called the magnetization current density, and

is the polarization current density. Taking the equation for

**B**:Consequently, referring to the definition of the bound current:

as was to be shown.

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

- ↑ Ampère never utilized the field concept in any of his works; cf. Assis, André Koch Torres; Chaib, J. P. M. C; Ampère, André-Marie (2015).
*Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience*(PDF). Montreal, QC: Apeiron. ch. 15 p. 221. ISBN 978-1-987980-03-5. The "Ampère circuital law" is thus more properly termed the "Ampère–Maxwell law." It is named after Ampère because of his contributions to understanding electric current. Maxwell does not take Ampère's force law as a starting point in deriving any of his equations, although he mentions Ampère's force law in his*A Treatise on Electricity and Magnetism*vol. 2, part 4, ch. 2 (§§502-527) & 23 (§§845-866). - ↑ Clerk Maxwell, James. "On Physical Lines of Force".
- ↑ Clerk Maxwell, James. "On Faraday's Lines of Force".
- ↑ Knoepfel, Heinz E. (2000).
*Magnetic Fields: A comprehensive theoretical treatise for practical use*. Wiley. p. 4. ISBN 0-471-32205-9. - ↑ Owen, George E. (2003).
*Electromagnetic Theory*(Reprint of 1963 ed.). Courier-Dover Publications. p. 213. ISBN 0-486-42830-3. - 1 2 Jackson, John David (1999).
*Classical Electrodynamics*(3rd ed.). Wiley. p. 238. ISBN 0-471-30932-X. - ↑ Griffiths, David J. (1999).
*Introduction to Electrodynamics*(3rd ed.). Pearson/Addison-Wesley. pp. 322–323. ISBN 0-13-805326-X. - ↑ Owen, George E. (2003).
*Electromagnetic Theory*. Mineola, NY: Dover Publications. p. 285. ISBN 0-486-42830-3. - ↑ Billingham, J.; King, A. C. (2006).
*Wave Motion*. Cambridge University Press. p. 179. ISBN 0-521-63450-4. - ↑ Slater, J. C.; Frank, N. H. (1969).
*Electromagnetism*(Reprint of 1947 ed.). Courier Dover Publications. p. 83. ISBN 0-486-62263-0. - ↑ Siegel, Daniel M. (2003).
*Innovation in Maxwell's Electromagnetic Theory: Molecular Vortices, Displacement Current, and Light*. Cambridge University Press. pp. 96–98. ISBN 0-521-53329-5. - ↑ Clerk Maxwell, James (1861). "On Physical Lines of Force" (PDF).
*Philosophical Magazine and Journal of Science*. - ↑ For example, see Griffiths, David J. (1999).
*Introduction to Electrodynamics*. Upper Saddle River, NJ: Prentice Hall. p. 323. ISBN 0-13-805326-X. and Tai L. Chow (2006).*Introduction to Electromagnetic Theory*. Jones & Bartlett. p. 204. ISBN 0-7637-3827-1. - ↑ Rogalski, Mircea S.; Palmer, Stuart B. (2006).
*Advanced University Physics*. CRC Press. p. 267. ISBN 1-58488-511-4. - ↑ Rogalski, Mircea S.; Palmer, Stuart B. (2006).
*Advanced University Physics*. CRC Press. p. 251. ISBN 1-58488-511-4. - ↑ The magnetization current can be expressed as the
*curl*of the magnetization, so its divergence is zero and it does not contribute to the continuity equation. See magnetization current.

- Griffiths, David J. (1998).
*Introduction to Electrodynamics (3rd ed.)*. Prentice Hall. ISBN 0-13-805326-X. - Tipler, Paul (2004).
*Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)*. W. H. Freeman. ISBN 0-7167-0810-8.

- 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

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge *q* moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

**Maxwell's equations** are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (*c*) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who published an early form of the equations that included the Lorentz force law between 1861 and 1862. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

A **magnetic field** is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

In electromagnetism, **displacement current density** is the quantity ∂* D*/∂

In classical electromagnetism, **polarization density** is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.

"**A Dynamical Theory of the Electromagnetic Field**" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

The term **magnetic potential** can be used for either of two quantities in classical electromagnetism: the *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential**ψ*. Both quantities can be used in certain circumstances to calculate the magnetic field **B**.

In physics, the **electric displacement field** or **electric induction** is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "**D**" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m^{−2}).

**Magnetostatics** is the study of magnetic fields in systems where the currents are steady. It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static — as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.

In electromagnetism, the **Lorenz gauge condition** or **Lorenz gauge** is a partial gauge fixing of the electromagnetic vector potential. The condition is that This does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function.

In classical electromagnetism, **magnetization** or **magnetic polarization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field. Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be *magnetized* to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector **M**.

**Lorentz–Heaviside units** constitute a system of units within CGS, named from Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant *ε*_{0} and magnetic constant *µ*_{0} do not appear, having been incorporated implicitly into the unit system and electromagnetic equations. Lorentz–Heaviside units may be regarded as normalizing *ε*_{0} = 1 and *µ*_{0} = 1, while at the same time revising Maxwell's equations to use the speed of light *c* instead.

**Micromagnetics** is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored, yet small enough to resolve magnetic structures such as domain walls or vortices.

The **covariant formulation of classical electromagnetism** refers to ways of writing the laws of classical electromagnetism in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.

In electromagnetism and applications, an **inhomogeneous electromagnetic wave equation**, or **nonhomogeneous electromagnetic wave equation**, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations *inhomogeneous*, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

In electromagnetism, **current density** is the amount of charge per unit time that flows through a unit area of a chosen cross section. The **current density vector** is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In electrodynamics, the **retarded potentials** are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light *c*, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution to another point in space, see figure below.

In physics, **Rosser's equation** aids in understanding the role of displacement current in Maxwell's equations, given that there is no aether in empty space as initially assumed by Maxwell. Due originally to William G.V. Rosser, the equation was labeled by Selvan:

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.