Maxwell's equations
Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
Introduction
Maxwell's four equations express, respectively, how electric charges produce electric fields (Gauss's law), the experimental absence of magnetic chargess, how currentss produce magnetic fields (Ampere's law), and how changing magnetic fields produce electric fields (Faraday's law of induction). Maxwell, in 1864, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields.
Furthermore, Maxwell showed that the four equations, with his correction, predict wavess of oscillating electric and magnetic fields that travel through empty space at a speed that could be predicted from simple electrical experiments—using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s. Maxwell (1865) wrote:
- This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.
Historical developments of Maxwell's equations and relativity
The modern mathematical formulation of Maxwell's equations is due to Oliver Heaviside and Willard Gibbs, who in 1884 reformulated Maxwell's original system of equations to a far simpler representation using vector calculus. (Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, although in 1873 he attempted a quaternion formulation that ultimately proved unpopular.) The change to the vector notation produced a symmetric mathematical representation that reinforced the perception of physical symmetries between the various fields. This highly symmetric formulation would directly inspire later developments in fundamental physics.
In the late 19th century, because of the appearance of a velocity,
The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. (In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.)
Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces is an active area of research in particle physics.
Summary of the Equations
General Case (SI units)
Name | Partial Differential form |
Gauss's law: | |
Gauss's law for magnetism: | |
Faraday's law of induction: | |
Ampere's law + Maxwell's extension: |
B is the magnetic flux density (in units of tesla, T), also called the magnetic induction.
D is the electric displacement field (in units of C/m^{2}).
E is the electric field (in units of V/m),
H is the magnetic field strength (in units of A/m)
∇· is the divergence operator,
Note that the second equation is equivalent to the statement that magnetic monopoles do not exist.
It is important to note that Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material, below. (The microscopic Maxwell's equations, ignoring quantum effects, are simply those of vacuum — but one must include all atomic charges and so on, which is normally an intractable problem.)
In Linear Materials
In linear materials, the D and H fields are related to E and B by:
where:ε is the electrical permittivity
μ is the magnetic permeability
(This can actually be extended to handle nonlinear materials as well, by making ε and μ depend upon the field strength; see e.g. the Kerr and Pockels effects.)
In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to
More generally, ε and μ can be rank-2 tensors (3×3 matrices) describing birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some material dispersion by which ε and/or μ depend upon frequency (and causality constrains this dependence to obey the Kramers-Kronig relations).
In Vacuum, Without Charges or Currents
The vacuum is a linear, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε_{0}; and μ_{0}; (neglecting very slight nonlinearities due to quantum effects). If there is no current or electric charge present in the vacuum, we obtain the Maxwell's equations in free space:
Detail
Charge density and the electric field
where is the free electric charge density (in units of C/m^{3}), not including dipole charges bound in a material, and is the electric displacement field (in units of C/m^{2}). This equation corresponds to Coulomb's law for stationary charges in vacuum.The equivalent integral form (by the divergence theorem), also known as Gauss's Law, is:
In a linear material , is directly related to the electric field via a material-dependent constant called the permittivity, :
- .
Compare Poisson's equation.
The Structure of the Magnetic Field
is the magnetic flux density (in units of tesla, T), also called the magnetic induction.Equivalent integral form:
is the area of a differential square on the surface with an outward facing surface normal defining its direction.Note: like the electric field's integral form, this equation only works if the integral is done over a closed surface.
This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is the mathematical formulation of the assumption that there are no magnetic monopoles.
A Changing Magnetic Field and the Electric Field
Equivalent Integral Form:
- where
Note: some textbooks show the right hand sign of the Integral form with an N (representing the number of coils of wire that are around the edge of A) in front of the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.
Note the negative sign; it is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.
This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. This equation describes how electric motors and electric generators work.
This law corresponds to the Faraday's law of electromagnetic induction.
Note: Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would mean a reversal of polarity of magnetic fields (not inconsistent, but confusingly against convention).
The Source of the Magnetic Field
where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: J = ∫ρ_{q}vdV where v is a vector field called the drift velocity that describes the velocities of that charge carriers which have a density described by the scalar function ρ_{q}.In free space, the permeability μ is the permeability of free space, μ_{0}, which is defined to be exactly 4π×10^{-7} W/Am. Thus, in free space, the equation becomes:
Note: unless there is a capacitor or some other place where , the second term on the right hand side is generally negligible and ignored. Any time this applies, the integral form is known as Ampère's Law.
Maxwell's Equations in CGS units
The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimetre, gram, second), the equations take on a more symmetrical form, as follows:
References
- James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
- James Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vols. 1-2 (1891) (reprinted: Dover, New York NY, 1954; ISBN 0-486-60636-8 and ISBN 0-486-60637-6).
- John David Jackson, Classical Electrodynamics (Wiley, New York, 1998).
- Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985).
- Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).
- Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995) ISBN 0-262-69188-4.
- Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987).
- Fitzpatrick, Richard, "Lecture series: Relativity and electromagnetism". Advanced Classical Electromagnetism, PHY387K. University of Texas at Austin, Fall 1996.