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.
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.
The original circuital law can be written in several different forms, which are all ultimately equivalent:
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.
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 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.
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.
Integral form | Differential form | |
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Using B-field and total current | ||
Using H-field and free current |
There are a number of ambiguities in the above definitions that require clarification and a choice of convention.
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 a vector field's curl divergence 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 experimental tests actually show that
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. |
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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
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 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. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.
Flux describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.
In physics, specifically electromagnetism, the Biot–Savart law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.
In electromagnetism, displacement current density is the quantity ∂D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.
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.
In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution, through energy flux.
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, denoted by D, 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 SI, it is expressed in units of coulomb per metre squared (C⋅m^{−2}).
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, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies 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.
In electromagnetism, Jefimenko's equations give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for moving charges and currents. They are the general solutions to Maxwell's equations for any arbitrary distribution of charges and currents.
Lorentz–Heaviside units constitute a system of units within CGS, named for 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.
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 makes 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 electric current per unit area of 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 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 electromagnetism, Ørsted's law, also spelled Oersted's law, is the physical law stating that an electric current creates a magnetic field.