Maxwell's equations

Table of contents

1 Introduction 2 Summary 2.1 General Case 2.2 In Linear Media/Material 2.3 In a Vacuum 3 Detail 3.4 Charge Density and the Electric Field 3.5 The Structure of the Magnetic Field 3.6 A Changing Magnetic Field and the Electric Field 3.7 The Source of the Magnetic Field 4 A Final Note on Unit Systems 5 Formulation of Maxwell's equations in special relativity 6 Maxwell's equations in terms of differential forms 7 References 8 External links

Introduction

The modern mathematical formulation of Maxwell's equations is due to Oliver Heaviside and Willard Gibbs, who in 1884 reformulated Maxwell's original equations 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 he ultimately found awkward.) The change to the vector notation produced a symmetric mathematical representation that reinforced the perception of physical symmetries between the various fields.

In the late 19th century, because of the appearance of a velocity () in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the Luminiferous aether (the postulated medium for light, whose interpretation was considerably debated). When the Michelson-Morley experiment conducted by Edward Morley and Albert Abraham Michelson produced a null result for the change of the velocity of light due to the Earth's hypothesized motion through the aether, however, alternative explanations were sought by Lorentz and others. This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute rest frame (or aether) and the invariance of Maxwell's equations in all frames of reference.

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.)

Kaluza and Klein showed in the 1920's that the 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

General Case

It is important to note that these equations are generally considered in terms of 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 continuous quantitites such as the permittivity or 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 Media/Material

In linear media, the macroscopic field strengths D and H are related to the bare field strengths E and B by

Detail

Charge Density and the Electric Field

,

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, :

.

The Structure of the Magnetic Field

is the magnetic flux density (in units of tesla, T), also called the magnetic induction.

is the area of a differential square on the surface  with an outward facing surface normal defining its direction.

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. This implies that there are no magnetic monopoles.

A Changing Magnetic Field and the Electric Field

Note: this equation only works of the surface A is not closed because the net magnetic flux through a closed surface will always be zero, as stated by the previous equation. That, and the electromotive force is measured along the edge of the surface; a closed surface has no edge. Some textbooks list 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

In free space, the permeability μ is the permeability of free space, μ₀, 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 Ampere's Law.

A Final Note on Unit Systems

The above equations are all in a unit system called mks (short for meter, kilogram, second; also know as the International System of Units (or SI for short). This is more commonly known as the metric system. In a related unit system, called cgs (short for centimeter, gram, second), the equations take on a more symmetrical form, as follows:

See also natural units, Lorentz-Heaviside units.

Formulation of Maxwell's equations in special relativity

to do: 4-vectors, and the d'Alembertian operator

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably once you use the language of differential geometry and differential forms. Now, the electric and magnetic fields are jointly described by a 2-form in a 4-dimensional spacetime manifold which is usually called F. Maxwell's equations then reduce to the Bianchi identity

References

• James Clerk Maxwell, "A dynamical theory of the electromagnetic field," Philosophical Transactions of the Royal Society of London 155, 459-512 (1865).
• 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).

• A tensor treatment of Maxwell's equations
• Lecture series: Relativity and electromagnetism