Ampère's circuital law

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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.

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In magnetostatics, the force of attraction or repulsion between two current-carrying wires is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, following the Biot–Savart law, and the other wire experiences a magnetic force as a consequence, following the Lorentz force law.

André-Marie Ampère was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as "electrodynamics". He is also the inventor of numerous applications, such as the solenoid and the electrical telegraph. An autodidact, Ampère was a member of the French Academy of Sciences and professor at the École polytechnique and the Collège de France.

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" [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).

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.

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 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 where μ0 is the magnetic constant.

The Kelvin–Stokes theorem, named after Lord Kelvin and George Stokes, also known as the Stokes' theorem, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface.

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often 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. Magnetic fields 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 varies with location. As such, it is an example of a vector field.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or forces to strains or deformations.

Explanation

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 mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

An electric current is a flow of electric charge. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

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 Ienc 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 If,enc through a surface S.

The tesla is a derived unit of the magnetic induction in the International System of Units.

The ampere, often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.

The metre or meter is the base unit of length in the International System of Units (SI). The SI unit symbol is m. The metre is defined as the length of the path travelled by light in a vacuum in 1/299 792 458 second.

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),
• Jf 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.

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 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 Ienc 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; it can be proven that any surface with boundary C can be chosen.

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 JM 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 JP.

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

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

with Jf 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.

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 a vector field's curl divergence must always be zero. Hence

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

and so the original Ampère's circuital law implies that

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

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

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

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. [6] [7] [8] [9] [10]

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

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

The circuital law implies that

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

but experimental tests actually show that

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

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

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 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]

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

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, εr 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. [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.

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 H-field is (the H-field is used because it includes the magnetization currents, so JM does not appear explicitly, see H-field and also Note): [14]

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

(integral form), where 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 Jf is the enclosed conduction current or free current density. In differential form,

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

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):

 ${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {l}}=\iint _{S}\left(\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} }$

In differential form,

 ${\displaystyle \mathbf {\nabla } \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$

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:

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

where current density JD 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 ε0E/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.

Ampère's circuital law in cgs units

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

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

where c is the speed of light.

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

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

Notes

1. 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).
2. Clerk Maxwell, James. "On Physical Lines of Force".
3. Clerk Maxwell, James. "On Faraday's Lines of Force".
4. Knoepfel, Heinz E. (2000). Magnetic Fields: A comprehensive theoretical treatise for practical use. Wiley. p. 4. ISBN   0-471-32205-9.
5. Owen, George E. (2003). Electromagnetic Theory (Reprint of 1963 ed.). Courier-Dover Publications. p. 213. ISBN   0-486-42830-3.
6. Jackson, John David (1999). Classical Electrodynamics (3rd ed.). Wiley. p. 238. ISBN   0-471-30932-X.
7. Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed.). Pearson/Addison-Wesley. pp. 322–323. ISBN   0-13-805326-X.
8. Owen, George E. (2003). Electromagnetic Theory. Mineola, NY: Dover Publications. p. 285. ISBN   0-486-42830-3.
9. Billingham, J.; King, A. C. (2006). Wave Motion. Cambridge University Press. p. 179. ISBN   0-521-63450-4.
10. Slater, J. C.; Frank, N. H. (1969). Electromagnetism (Reprint of 1947 ed.). Courier Dover Publications. p. 83. ISBN   0-486-62263-0.
11. 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.
12. Clerk Maxwell, James (1861). "On Physical Lines of Force" (PDF). Philosophical Magazine and Journal of Science.
13. 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.
14. Rogalski, Mircea S.; Palmer, Stuart B. (2006). Advanced University Physics. CRC Press. p. 267. ISBN   1-58488-511-4.
15. Rogalski, Mircea S.; Palmer, Stuart B. (2006). Advanced University Physics. CRC Press. p. 251. ISBN   1-58488-511-4.
16. 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.