Maxwells equations Wikipedia. Maxwells equations mid left as featured on a monument in front of Warsaw Universitys Center of New Technologies. Maxwells equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, quantum field theory, classical optics, and electric circuits. They underpin all electric, optical and radio technologies, including power generation, electric motors, wireless communication, cameras, televisions, computers etc. Maxwells equations describe how electric and magnetic fields are generated by charges, currents, and changes of each other. 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 rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1. The equations have two major variants. The microscopic Maxwell 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 Maxwell equations define two new auxiliary fields that describe the large scale behaviour of matter without having to consider atomic scale details. However, their use requires experimentally determining parameters for a phenomenological description of the electromagnetic response of materials. A system of linear equations or linear system is a collection of linear equations involving the same set of variables. For example. OR. Linear Algebra with Applications, 3rd Edition by Otto Bretscher ISBN 0131453343 ISBN 13 9780131453340 Publisher Prentice Hall. Download Abstract Algebra Theory and Applications PDF 442P Download free online book chm pdf. Maxwells equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, quantum. The term Maxwells equations is often used for equivalent alternative formulations. Versions of Maxwells equations based on the electric and magnetic potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
The spacetime formulations i. In fact, Einstein developed special and general relativity to accommodate the absolute speed of light that drops out of the Maxwell equations with the principle that only relative movement has physical consequences. Since the mid 2. Maxwells equations are not exact, but a classical field theory approximation of some aspects of the fundamental theory of quantum electrodynamics, although some quantum features, such as quantum entanglement, are completely absent and in no way approximated. For example, quantum cryptography has no approximate version in Maxwell theory. In many situations, though, deviations from Maxwells equations are immeasurably small. Exceptions include nonclassical light, photonphoton scattering, quantum optics, and many other phenomena related to photons or virtual photons. Formulation in terms of electric and magnetic fields microscopic or in vacuum versioneditIn the electric and magnetic field formulation there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources. Gausss law describes how electric fields emanate from electric charges. Gausss law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles. The two homogeneous equations describe how the fields circulate around their respective sources. Ampres law with Maxwells addition describes how the magnetic field circulates around electric currents and time varying electric fields, while Faradays law describes how the electric field circulates around time varying magnetic fields. A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is no longer. The precise formulation of Maxwells equations depends on the precise definition of the quantities involved. Conventions differ with the unit systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of lightc. This makes constants come out differently. The most common form is based on conventions used when quantities measured using SI units, but other commonly used conventions are used with other units including Gaussian units based on the cgs system,1LorentzHeaviside units used mainly in particle physics, and Planck units used in theoretical physics. The vector calculus formulation below has become standard. It is mathematically much more convenient than Maxwells original 2. Oliver Heaviside. 23 The differential and integral equations formulations are mathematically equivalent and are both 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. 4 For formulations using tensor calculus or differential forms, see alternative formulations. For relativistically invariant formulations, see relativistic formulations. Formulation in SI units conventioneditFormulation in Gaussian units conventioneditGaussian units are a popular system of units that are part of the centimetregramsecond system of units cgs. When using Gaussian units it is conventional to use a slightly different definition of electric field Ecgs c1. ESI. This implies that the modified electric and magnetic field have the same units in the SI convention this is not the case making dimensional analysis of the equations different e. ESIcSIBSIdisplaystyle mathbf E mathrm SI cmathrm SI mathbf B mathrm SI. The Gaussian system uses a unit of charge defined in such a way that the permittivity of the vacuum 0 14c, hence 0 4c. These units are sometimes preferred over SI units in the context of special relativity,5 vii in which the components of the electromagnetic tensor, the Lorentz covariant object describing the electromagnetic field, have the same unit without constant factors. Using these different conventions, the Maxwell equations become 6Key to the notationeditSymbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence. The sources are. The universal constants appearing in the equations are. Differential equationseditIn the differential equations,Integral equationseditIn the integral equations, is any fixed volume with closed boundary surface, and is any fixed surface with closed boundary curve ,Here a fixed volume or surface means that it does not change over time. The equations are correct, complete and a little easier to interpret with time independent surfaces. For example, since the surface is time independent, we can bring the differentiation under the integral sign in Faradays law ddtBd. SBtd. S,displaystyle frac ddtiint Sigma mathbf B cdot mathrm d mathbf S iint Sigma frac partial mathbf B partial tcdot mathrm d mathbf S ,Maxwells equations can be formulated with possibly time dependent surfaces and volumes by substituting the left hand side with the right hand side in the integral equation version of the Maxwell equations. Q d. V,displaystyle Qiiint Omega rho mathrm d V,where d. V is the volume element. IJd. S,displaystyle Iiint Sigma mathbf J cdot mathrm d mathbf S ,where d. S denotes the vector element of surface area S, normal to surface. Vector area is sometimes denoted by A rather than S, but this conflicts with the notation for magnetic potential. Relationship between differential and integral formulationseditThe equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the KelvinStokes theorem. Flux and divergenceedit. Volume and its closed boundary, containing respectively enclosing a source and sink of a vector field F. Here, F could be the E field with source electric charges, but not the B field which has no magnetic charges as shown.