Maxwell's equations

Maxwell's equations are the set of four equations by James Clerk Maxwell that describe the behavior of both the electric and magnetic fields. Maxwell's equations provided the basis for the unification of electric field and magnetic field, the electromagnetic description of light, and ultimately, Albert Einstein's theory of relativity.

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, he later attempted a quaternion formulation). 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, 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 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.)

The Equations

Charge Density and the Electric Field

<math>\nabla \cdot \mathbf{D} = \rho</math>

where <math>{\rho}</math> is the electric charge density (in units of C/m³), and <math>\mathbf{D}</math> is the electric displacement field[?] (in units of C/m²) which in a linear material is related to the electric field <math>\mathbf{E}</math> via a material-dependent constant called the permittivity, <math>\epsilon</math>: <math>\mathbf{D} = \epsilon\mathbf{E}</math>. Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as <math>\epsilon_0</math>, resulting in the equation for free space:

<math>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}</math>

where, again, <math>\mathbf{E}</math> is the electric field (in units of V/m), <math>\rho</math> is the charge density, and <math>\epsilon_0</math> (approximately 8.854 pF/m) is the permittivity of free space. In a linear material, however, <math>\epsilon_0</math> would be replaced with <math>\epsilon</math>, where <math>\epsilon = \epsilon_0 \times \epsilon_r</math>, and <math>\epsilon_r</math> is the material's relative dielectric constant.

Equivalent integral form:

<math>\int_A \mathbf{E} \cdot d\mathbf{A} = \frac{Q_\mbox{enclosed}}{\epsilon_0}</math>

where <math>d\mathbf{A}</math> is the area of a differential square on the surface A with an outward facing surface normal defining its direction, <math>Q_\mbox{enclosed}</math> is the charge enclosed by the surface.

Note: the integral form only works if the integral is over a closed surface. Shape and size do not matter. The integral form is also known as Gauss's Law.

This equation corresponds to Coulomb's law for stationary charges.

The Structure of the Magnetic Field

<math>\nabla \cdot \mathbf{B} = 0</math>

<math>\mathbf{B}</math> is the magnetic flux density (in units of tesla, T), also called the magnetic induction.

Equivalent integral form:

<math>\int_A \mathbf{B} \cdot d\mathbf{A} = 0</math>

<math>d\mathbf{A}</math> is the area of a differential square on the surface <math>A</math> 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. This implies that there are no magnetic monopoles. If a monopole were to be discovered, this equation would need to be modified to read

<math>\nabla \cdot \mathbf{B} = \rho_m</math>

where <math>\rho_m</math> would be the density of magnetic monopoles.

A Changing Magnetic Field and the Electric Field

<math>\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}</math>

Equivalent Integral Form:

<math> \epsilon = - \frac {d\Phi_{\mathbf{B}}} {dt}</math> where <math> \Phi_{\mathbf{B}} = \int_{A} \mathbf{B} \cdot d\mathbf{A}</math>

Φ_B is the magnetic flux through the area A described by the second equation, ε is the electromotive force around the edge of the surface A.

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

<math> \nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}} {\partial t}</math>

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_qvdV 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, μ₀, which is defined to be exactly 4π×10^-7 W/Am. Thus, in free space, the equation becomes:

<math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{D}}{\partial t}</math>

Equivalent integral form:

<math>\int_s \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_\mbox{encircled} - \mu_0 \epsilon_0 \int_A \frac{\partial \mathbf{D}}{\partial t} \cdot d \mathbf{A}</math>

s is the edge of the open surface A (any surface with the curve s as its edge will do), and I_encircled is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_AJ·dA).

Note: unless there is a capacitor or some other place where <math>\nabla \cdot \mathbf{J} \ne 0</math>, the second term on the right hand side is generally negligable and ignored. Any time this applies, the integral form is known as Ampere's Law.

Summary

<math>\nabla \cdot \mathbf{D} = \rho </math>

<math>\nabla \cdot \mathbf{B} = 0</math>

<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>

<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>

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

<math>D = \epsilon E</math>

<math>B = \mu H</math>

In linear and isotropic media, ε and μ are constants, and Maxwell's equations reduce to

<math>\nabla \cdot \mathbf{E} = \frac{\rho} {\epsilon}</math>

<math>\nabla \cdot \mathbf{B} = 0</math>

<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>

<math>\nabla \times \mathbf{B} = \mu \mathbf{J} + \mu \epsilon \frac{\partial \mathbf{E}} {\partial t}</math>

The vacuum is such a medium, and the proportionality constants in the vacuum are denoted by ε₀ and μ₀. If there is no current or electric charge present in the vacuum, we obtain the Maxwell equation's in free space:

<math>\nabla \cdot \mathbf{E} = 0</math>

<math>\nabla \cdot \mathbf{B} = 0</math>

<math>\nabla \times \mathbf{E} = -\frac{\partial\mathbf{B}} {\partial t}</math>

<math>\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t}</math>

This equation has a simple solution in terms of travelling sinusoidal plane waves, with the elecric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields 90° out of phase, travelling at the speed

<math>c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} </math>

Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation.

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:

<math> \nabla \cdot \mathbf{E} = 4\pi\rho</math>

<math> \nabla \cdot \mathbf{B} = 0</math>

<math> \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>

<math> \nabla \times \mathbf{B} = \frac{1}{c} \frac{ \partial \mathbf{E}} {\partial t} + \frac{4\pi}{c} \mathbf{J}</math>

Where c is the speed of light in a vacuum. The symmetry is more apparent when the electromagnetic field is considered in a vacuum. The equations take on the following, highly symmetric form:

<math>\nabla \cdot \mathbf{E} = 0</math>

<math>\nabla \cdot \mathbf{B} = 0</math>

<math>\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}</math>

<math>\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} </math>

Note: All variables that are in bold represent vector quantities; see also vector calculus.

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).