# Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862. The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability but may be unfeasible to calculate. The term "Maxwell's equations" is often used for other forms of Maxwell's equations. Formulation in terms of electric and magnetic fields Flux and divergence

THE DISPLACEMENT CURRENT AND MAXWELLS EQUATIONS 35.1. The displacement current The calculation of the magnetic field of a current distribution can, in principle, be carried out using Ampere's law which relates the path integral of the magnetic field around a closed path to the current intercepted by an arbitrary surface that spans this path: Ampere's law is independent of the shape of the surface chosen as long as the current flows along a continuous, unbroken circuit. Figure 35.1. Although the surface shown in Figure 35.1 does not intercept any current, it intercepts electric flux. The electric field outside the capacitor is equal to zero. If a current I is flowing through the wire, then the charge on the capacitor plates will be time dependent. The magnetic field around the wire can now be found by modifying Ampere's law where [Phi]E is the electric flux through the surface indicated in Figure 35.1 In the most general case, the surface spanned by the integration path of the magnetic field can intercept current and electric flux. Thus

Weber (unit) The weber is named after the German physicist Wilhelm Eduard Weber (1804–1891). The weber may be defined in terms of Faraday's law, which relates a changing magnetic flux through a loop to the electric field around the loop. A change in flux of one weber per second will induce an electromotive force of one volt (produce an electric potential difference of one volt across two open-circuited terminals). Officially, Weber (unit of magnetic flux) — The weber is the magnetic flux which, linking a circuit of one turn, would produce in it an electromotive force of 1 volt if it were reduced to zero at a uniform rate in 1 second.[2] The weber is commonly expressed in a multitude of other units: This SI unit is named after Wilhelm Eduard Weber. It was not until 1927 that TC1 dealt with the study of various outstanding problems concerning electrical and magnetic quantities and units. Also in 1935, TC1 passed responsibility for "electric and magnetic magnitudes and units" to the new TC24.

A Dynamical Theory of the Electromagnetic Field "A Dynamical Theory of the Electromagnetic Field" is the third of James Clerk Maxwell's papers regarding electromagnetism, published in 1865.[1] It is the paper in which the original set of four Maxwell's equations first appeared. The concept of displacement current, which he had introduced in his 1861 paper "On Physical Lines of Force", was utilized for the first time, to derive the electromagnetic wave equation.[2] Maxwell's original equations In part III of "A Dynamical Theory of the Electromagnetic Field", which is entitled "General Equations of the Electromagnetic Field", Maxwell formulated twenty equations[1] which were to become known as Maxwell's equations, until this term became applied instead to a vectorized set of four equations selected in 1884, which had all appeared in "On physical lines of force".[2] Heaviside's versions of Maxwell's equations are distinct by virtue of the fact that they are written in modern vector notation. For his original text on force, see: .

Inductance In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current flowing through it induces an electromotive force in both the conductor itself[1] and in any nearby conductors by mutual inductance.[1] These effects are derived from two fundamental observations of physics: a steady current creates a steady magnetic field described by Oersted's law,[2] and a time-varying magnetic field induces an electromotive force in nearby conductors, which is described by Faraday's law of induction.[3] According to Lenz's law,[4] a changing electric current through a circuit that contains inductance induces a proportional voltage, which opposes the change in current (self-inductance). The varying field in this circuit may also induce an e.m.f. in neighbouring circuits (mutual inductance). Circuit analysis An electronic component that is intended to add inductance to a circuit is called an inductor. Here, inductance L is a symmetric matrix. and

Covariant formulation of classical electromagnetism The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems. This article uses SI units for the purely spatial components of tensors (including vectors), the classical treatment of tensors and the Einstein summation convention throughout, and the Minkowski metric has the form diag (+1, −1, −1, −1). Covariant objects Preliminary 4-vectors In meter−1 the four-gradient is

Moving magnet and conductor problem Conductor moving in a magnetic field. The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. This problem, along with the Fizeau experiment, the aberration of light, and more indirectly the negative aether drift tests such as the Michelson–Morley experiment, formed the basis of Einstein's development of the theory of relativity.[2] Introduction Einstein's 1905 paper that introduced the world to relativity opens with a description of the magnet/conductor problem.[1] It is known that Maxwell's electrodynamics – as usually understood at the present time – when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. where .

Aharonov–Bohm effect Werner Ehrenberg and Raymond E. Siday first predicted the effect in 1949,[3] and similar effects were later published by Yakir Aharonov and David Bohm in 1959.[4] After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper.[5][6] Subsequently, the effect was confirmed experimentally by several authors; a general review can be found in Peshkin and Tonomura (1989).[7] Significance The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of (Maxwell's) classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical. The three issues are: Potentials vs. fields Magnetic solenoid effect .

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