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
Faraday paradoxThe Faraday paradox (or Faraday's paradox) is any experiment in which Michael Faraday's law of electromagnetic induction appears to predict an incorrect result. The paradoxes fall into two classes: 1. Faraday's law predicts that there will be zero EMF but there is a non-zero EMF. 2. Faraday's law predicts that there will be a non-zero EMF but there is a zero EMF. Faraday deduced this law in 1831, after inventing the first electromagnetic generator or dynamo, but was never satisfied with his own explanation of the paradox. Paradoxes in which Faraday's law of induction predicts zero EMF but there is a non-zero EMF These paradoxes are generally resolved by the fact that an EMF may be created by a changing flux in a circuit as explained in Faraday's law or by the movement of a conductor in a magnetic field. The equipment Figure 1: Faraday's disc electric generator. The procedure The experiment proceeds in three steps: Why is this paradoxical? Faraday's explanation we get:
Kennedy–Thorndike experimentFigure 1. The Kennedy–Thorndike experiment Improved variants of the Kennedy–Thorndike experiment have been conducted using optical cavities or Lunar Laser Ranging. For a general overview of tests of Lorentz invariance, see Tests of special relativity. The experiment The original Michelson–Morley experiment was useful for testing the Lorentz–FitzGerald contraction hypothesis only. The principle on which this experiment is based is the simple proposition that if a beam of homogeneous light is split […] into two beams which after traversing paths of different lengths are brought together again, then the relative phases […] will depend […] on the velocity of the apparatus unless the frequency of the light depends […] on the velocity in the way required by relativity. Referring to Fig. 1, key optical components were mounted within vacuum chamber V on a fused quartz base of extremely low coefficient of thermal expansion. Theory Basic theory of the experiment Figure 2. where
Dirac equationIn particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles, for which parity is a symmetry, such as electrons and quarks, and is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represent one of the great triumphs of theoretical physics. Mathematical formulation The Dirac equation in the form originally proposed by Dirac is: where ψ = ψ(x, t) is the wave function for the electron of rest mass m with spacetime coordinates x, t. Dirac's coup with
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. 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.
Time dilation of moving particlesRelation between the Lorentz factor γ and the time dilation of moving clocks. Time dilation of moving particles as predicted by special relativity can be measured in particle lifetime experiments. According to special relativity, the rate of clock C traveling between two synchronized laboratory clocks A and B is slowed with respect to the laboratory clock rates. This effect is called time dilation. Since any periodic process can be considered a clock, also the lifetimes of unstable particles such as muons must be affected, so that moving muons should have a longer lifetime than resting ones. Atmospheric tests a) View in S b) View in S′ c) Loedel diagram (In order to make the differences smaller, 0.7c was used instead of 0.995c) Theory The emergence of the muons is caused by the collision of cosmic rays with the upper atmosphere, after which the muons reach Earth. Time dilation and length contraction Length of the atmosphere: The contraction formula is given by Minkowski diagram and
Fourier series, wikipediaIn mathematics, a Fourier series (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The Discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. And the Z-transform reduces to a Fourier series for the important case |z|=1. Fourier series is also central to the original proof of the Nyquist–Shannon sampling theorem. History The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. The heat equation is a partial differential equation. Definition is a periodic function with period P. we can also write the function in these equivalent forms: where: approximates on Other applications .
InductanceIn 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 and in any nearby conductors by mutual inductance. These effects are derived from two fundamental observations of physics: a steady current creates a steady magnetic field described by Oersted's law, and a time-varying magnetic field induces an electromotive force in nearby conductors, which is described by Faraday's law of induction. According to Lenz's law, 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
Relativistic Doppler effectDiagram 1. A source of light waves moving to the right, relative to observers, with velocity 0.7c. The frequency is higher for observers on the right, and lower for observers on the left. The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity. The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. Visualization In Diagram 2, the blue point represents the observer, and the arrow represents the observer's velocity vector. Diagram 3. Analogy Understanding relativistic Doppler effect requires understanding the Doppler effect, time dilation, and the aberration of light. Motion along the line of sight where