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Maxwell's Equations - Wikipedia, the Free Encyclopedia Page 1 of 35 Maxwell's equations - Wikipedia, the free encyclopedia Page 1 of 35 Maxwell's equations From Wikipedia, the free encyclopedia (Redirected from Electromagnetic theory) Maxwell's equations are a set of partial differential equations Electromagnetism that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These in turn underlie modern electrical and communications technologies. Maxwell's equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current Electricity · Magnetism including the difficult-to-calculate atomic level charges and Electrostatics currents in materials. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that can sidestep Electric charge · Coulomb's law · having to know these 'atomic' sized charges and currents. Electric field · Electric flux · Gauss's law · Electric potential · Maxwell's equations are named after the Scottish physicist and Electrostatic induction · mathematician James Clerk Maxwell, since in an early form they Electric dipole moment · are all found in a four-part paper, "On Physical Lines of Force," which he published between 1861 and 1862. The mathematical Polarization density form of the Lorentz force law also appeared in this paper. Magnetostatics It is often useful to write Maxwell's equations in other forms; Ampère's law · Electric current · these representations are still formally termed "Maxwell's Magnetic field · Magnetization · equations". A relativistic formulation in terms of covariant field Magnetic flux · Biot–Savart law · tensors is used in special relativity, while, in quantum mechanics, Magnetic dipole moment · a version based on the electric and magnetic potentials is preferred. Gauss's law for magnetism Electrodynamics Lorentz force law · emf · Contents Electromagnetic induction · Faraday’s law · Lenz's law · ■ 1 Conceptual description Displacement current · ■ 1.1 Gauss's law Maxwell's equations · EM field · ■ 1.2 Gauss's law for magnetism Electromagnetic radiation · ■ 1.3 Faraday's law ■ 1.4 Ampère's law with Maxwell's correction Liénard–Wiechert potential · Maxwell tensor · Eddy current ■ 2 Units and summary of equations ■ 2.1 Table of 'microscopic' equations Electrical Network ■ 2.2 Table of 'macroscopic' equations Electrical conduction · ■ 2.3 Table of terms used in Maxwell's equations ■ 2.4 Proof that the two general formulations are Electrical resistance · Capacitance · equivalent Inductance · Impedance · Resonant cavities · Waveguides ■ 3 Maxwell's 'microscopic' equations ■ 3.1 With neither charges nor currents Covariant formulation ■ 4 Maxwell's 'macroscopic' equations ■ 4.1 Bound charge and current http://en.wikipedia.org/wiki/Electromagnetic_theory 5/31/2011 Maxwell's equations - Wikipedia, the free encyclopedia Page 2 of 35 ■ 4.2 Equations Electromagnetic tensor · ■ 4.3 Constitutive relations EM Stress-energy tensor · ■ 4.3.1 Without magnetic or dielectric materials Four-current · ■ 4.3.2 Isotropic Linear materials Electromagnetic four-potential ■ 4.3.3 General case Scientists ■ 4.3.4 Calculation of constitutive relations Ampère · Coulomb · Faraday · ■ 5 History ■ 5.1 Relation between electricity, magnetism, and Gauss · Heaviside · Henry · Hertz · the speed of light Lorentz · Maxwell · Tesla · Volta · ■ 5.2 The term Maxwell's equations Weber · Ørsted ■ 5.3 On Physical Lines of Force ■ 5.4 A Dynamical Theory of the Electromagnetic Field ■ 5.5 A Treatise on Electricity and Magnetism ■ 5.6 Maxwell's equations and relativity ■ 6 Modified to include magnetic monopoles ■ 7 Boundary conditions using Maxwell's equations ■ 8 Gaussian units ■ 9 Alternative formulations of Maxwell's equations ■ 9.1 Covariant formulation of Maxwell's equations ■ 9.2 Potential formulation ■ 9.3 Four-potential ■ 9.4 Differential geometric formulations ■ 9.4.1 Conceptual insight from this formulation ■ 9.5 Geometric Algebra (GA) formulation ■ 10 Classical electrodynamics as the curvature of a line bundle ■ 11 Curved spacetime ■ 11.1 Traditional formulation ■ 11.2 Formulation in terms of differential forms ■ 12 See also ■ 13 Notes ■ 14 References ■ 15 Further reading ■ 15.1 Journal articles ■ 15.2 University level textbooks ■ 15.2.1 Undergraduate ■ 15.2.2 Graduate ■ 15.2.3 Older classics ■ 15.2.4 Computational techniques ■ 16 External links ■ 16.1 Modern treatments ■ 16.2 Historical ■ 16.3 Other http://en.wikipedia.org/wiki/Electromagnetic_theory 5/31/2011 Maxwell's equations - Wikipedia, the free encyclopedia Page 3 of 35 Conceptual description Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. (See below for a mathematical description of these laws.) Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations describe how the fields 'circulate' around their respective sources; the magnetic field 'circulates' around electric currents and time varying electric field in Ampère's law with Maxwell's correction, while the electric field 'circulates' around time varying magnetic fields in Faraday's law. Gauss's law Main article: Gauss's law Gauss's law describes the relationship between an electric field and the generating electric charges: The electric field points away from positive charges and towards negative charges. In the field line description, electric field lines begin only at positive electric charges and end only at negative electric charges. 'Counting' the number of field lines in a closed surface, therefore, yields the total charge enclosed by that surface. More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the enclosed electric charge. Gauss's law for magnetism Main article: Gauss's law for magnetism Gauss's law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges. [1] Instead, the magnetic field due to materials is generated by a configuration called a dipole. Magnetic dipoles are best represented as loops of current but resemble positive and negative 'magnetic charges', inseparably bound together, having no net 'magnetic charge'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must somewhere Gauss's law for magnetism: magnetic exit that volume. Equivalent technical statements are that the field lines never begin nor end but sum total magnetic flux through any Gaussian surface is zero, or form loops or extend to infinity as that the magnetic field is a solenoidal vector field. shown here with the magnetic field due to a ring of current. Faraday's law Main article: Faraday's law Faraday's law describes how a time varying magnetic field creates ("induces") an electric field.[1] This aspect of electromagnetic induction is the operating principle behind many electric generators: for example a rotating bar magnet creates a changing magnetic field, which in turn generates an electric http://en.wikipedia.org/wiki/Electromagnetic_theory 5/31/2011 Maxwell's equations - Wikipedia, the free encyclopedia Page 4 of 35 field in a nearby wire. (Note: there are two closely related equations which are called Faraday's law. The form used in Maxwell's equations is always valid but more restrictive than that originally formulated by Michael Faraday .) Ampère's law with Maxwell's correction Main article: Ampère's law with Maxwell's correction In a geomagnetic storm, a surge in the flux of charged particles temporarily Ampère's law with alters Earth's magnetic field, which Maxwell's correction induces electric fields in Earth's states that magnetic fields atmosphere, thus causing surges in can be generated in two our electrical power grids. ways: by electrical current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's correction"). Maxwell's correction to Ampère's law is particularly important: It means that a changing magnetic field creates an electric field, An Wang's magnetic core memory and a changing electric field creates a magnetic field. [1][2] (1954) is an application of Ampere's Therefore, these equations allow self-sustaining "electromagnetic law. Each core stores one bit of data. waves" to travel through empty space (see electromagnetic wave equation ). The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, [note 1] exactly 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 previously-separate fields of electromagnetism and optics . Units and summary of equations Maxwell's equations vary with the unit system used. Though the general form remains the same, various definitions get changed and different constants appear at different places. (This may seem strange at first, but this is because some unit systems, e.g. variants of cgs, define their units in such a way that certain physical constants
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