Maxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges and currents. Together with the Lorentz force law, they form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, Wireless communication, lenses, and radar.
Maxwell's equations have two major variants: The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.
The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity.In general relativity, however, they must enter, through its stress–energy tensor, into Einstein field equations that include the spacetime curvature.
Maxwell's equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.
Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.
Summary The two most commonly used variants of Maxwell's equations are the microscopic and macroscopic differential form in SI units.
Microscopic version in SI units In partial differential equation form and in SI units, Maxwell's microscopic equations are written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law)
In these equations, is the electric field, the magnetic field, the total electric charge density, the total current density, the vacuum permittivity, the vacuum permeability, the divergence operator, and the curl operator.
Macroscopic version in SI units
In partial differential equation form and in SI units, Maxwell's macroscopic equations are written as:
In these equations, is the electric displacement field, the electric field, the magnetic field, the magnetic field intensity, the free electric charge density, the free current density, the divergence operator, and the curl operator.
History of the equations
Conceptual descriptions Gauss's law
Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.
Gauss's law for magnetism
Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation. Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.The absence of sinks/sources of the field does not imply that the field lines must be closed or escape to infinity. They can also wrap around indefinitely, without self-intersections. Moreover, around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined), there can be the simultaneous begin of some lines and end of other lines. This happens, for instance, in the middle between two identical cylindrical magnets, whose north poles face each other. In the middle between those magnets, the field is zero and the axial field lines coming from the magnets end. At the same time, an infinite number of divergent lines emanate radially from this point. The simultaneous presence of lines which end and begin around the point preserves the divergence-free character of the field. For a detailed discussion of non-closed field lines, see L. Zilberti "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.
Faraday's law
The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.
The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field and generates an electric field in a nearby wire.
Ampère–Maxwell law
The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields.J. D. Jackson, Classical Electrodynamics, section 6.3 As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field.Principles of physics: a calculus-based text, by R. A. Serway, J. W. Jewett, page 809. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.
The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,The quantity we would now call , with units of velocity, was directly measured before Maxwell's equations, in an 1855 experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch. They charged a leyden jar (a kind of capacitor), and measured the electrostatic force associated with the potential; then, they discharged it while measuring the magnetic force from the current in the discharge wire. Their result was , remarkably close to the speed of light. See Joseph F. Keithley, The story of electrical and magnetic measurements: from 500 B.C. to the 1940s, p. 115. matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
Microscopic formulation in terms of electric and magnetic fields (in vacuum version)
In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside,Bruce J. Hunt (1991) The Maxwellians, chapter 5 and appendix, Cornell University Press has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x, y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see ).
The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis.
Key to the notation
Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field, , a vector field, and the magnetic field, , a pseudovector field, each generally having a time and location dependence. The sources are the total electric charge density (total charge per unit volume), , and the total electric current density (total current per unit area), .
The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are: the permittivity of free space, , and the permeability of free space, , and the speed of light,
Differential equations In the differential equations, the nabla symbol, , denotes the three-dimensional gradient operator, del, the symbol (pronounced "del dot") denotes the divergence operator, the symbol (pronounced "del cross") denotes the curl operator.
Integral equations In the integral equations, is any volume with closed boundary surface , and is any surface with closed boundary curve , The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:
Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate. is a surface integral over the boundary surface , with the loop indicating the surface is closed is a volume integral over the volume , is a line integral around the boundary curve , with the loop indicating the curve is closed. is a surface integral over the surface , The total electric charge enclosed in is the volume integral over of the charge density (see the "macroscopic formulation" section below): where is the volume element. The net magnetic flux is the surface integral of the magnetic field passing through a fixed surface, : The net electric flux is the surface integral of the electric field passing through : The net electric current is the surface integral of the electric current density passing through : where denotes the differential vector element of surface area , normal to surface . (Vector area is sometimes denoted by rather than , but this conflicts with the notation for magnetic vector potential).