Faraday's law of induction
Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force —a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers and many types of electrical motors and solenoids; the Maxwell–Faraday equation describes the fact that a spatially varying electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is EMF on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time. Faraday's law had been discovered and one aspect of it was formulated as the Maxwell–Faraday equation later. Interestingly, the equation of Faraday's law can be derived by the Maxwell–Faraday equation and the Lorentz force; the integral form of the Maxwell–Faraday equation describes only the transformer EMF, while the equation of Faraday's law describes both the transformer EMF and the motional EMF.
Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to publish the results of his experiments. In Faraday's first experimental demonstration of electromagnetic induction, he wrapped two wires around opposite sides of an iron ring. Based on his assessment of discovered properties of electromagnets, he expected that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side, he plugged one wire into a galvanometer, watched it as he connected the other wire to a battery. Indeed, he saw a transient current when he connected the wire to the battery, another when he disconnected it; this induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. Within two months, Faraday had found several other manifestations of electromagnetic induction. For example, he saw transient currents when he slid a bar magnet in and out of a coil of wires, he generated a steady current by rotating a copper disk near the bar magnet with a sliding electrical lead.
Michael Faraday explained electromagnetic induction using a concept. However, scientists at the time rejected his theoretical ideas because they were not formulated mathematically. An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory. In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law though it is different from the original version of Faraday's law, does not describe motional EMF. Heaviside's version is the form recognized today in the group of equations known as Maxwell's equations. Lenz's law, formulated by Emil Lenz in 1834, describes "flux through the circuit", gives the direction of the induced EMF and current resulting from electromagnetic induction; the most widespread version of Faraday's law states: The electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path.
The closed path here is, in fact, conductive. For a loop of wire in a magnetic field, the magnetic flux ΦB is defined for any surface Σ whose boundary is the given loop. Since the wire loop may be moving, we write Σ for the surface; the magnetic flux is the surface integral: Φ B = ∬ Σ B ⋅ d A, where dA is an element of surface area of the moving surface Σ, B is the magnetic field, B·dA is a vector dot product representing the element of flux through dA. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic flux lines that pass through the loop; when the flux changes—because B changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an EMF, E, defined as the energy available from a unit charge that has travelled once around the wire loop. Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, attaching a voltmeter to the leads. Faraday's law states that the EMF is given by the rate of change of the magnetic flux: E = − d Φ B d t, where E is the electromotive force and ΦB is the magnetic flux.
The direction of the electromotive force is given by Lenz's law. The laws of induction of electric currents in mathematical form was established by Franz Ernst Neumann in 18
Permittivity
In electromagnetism, absolute permittivity simply called permittivity denoted by the Greek letter ε, is the measure of capacitance, encountered when forming an electric field in a particular medium. More permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium; the SI unit for permittivity is farad per meter. The lowest possible permittivity is that of a vacuum. Vacuum permittivity, sometimes called the electric constant, is represented by ε0 and has a value of 8.85×10−12 F/m. The permittivity of a dielectric medium is represented by the ratio of its absolute permittivity to the electric constant; this dimensionless quantity is called the medium’s relative permittivity, sometimes called "permittivity". Relative permittivity is commonly referred to as the dielectric constant, a term, deprecated in physics and engineering as well as in chemistry.
Κ = ε r = ε ε 0 By definition, a perfect vacuum has a relative permittivity of 1. The difference in permittivity between a vacuum and air can be considered negligible, as κair = 1.0006. Relative permittivity is directly related to electric susceptibility, a measure of how a dielectric polarizes in response to an electric field, given by χ = κ − 1 otherwise written as ε = ε r ε 0 = ε 0 The standard SI unit for permittivity is Farad per meter. F m = C V ⋅ m = C 2 N ⋅ m 2 = A 2 ⋅ s 4 kg ⋅ m 3 = N V 2 In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electric charges in a given medium, including charge migration and electric dipole reorientation, its relation to permittivity in the simple case of linear, isotropic materials with "instantaneous" response to changes in electric field is D = ε E where the permittivity ε is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor. In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity and other parameters.
In a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on complex values. In SI units, permittivity is measured in farads per meter; the displacement field D is measured in units of coulombs per square meter, while the electric field E is measured in volts per meter. D and E describe the interaction between charged objects. D is related to the charge densities associated with this interaction, while E is related to the forces and potential differences; the vacuum permittivity ε0 is the ratio D/E in free space. It appears in the Coulomb force constant, k e = 1 4 π ε 0 Its value is ε 0 = d e f 1 c 0 2 μ 0 = 1 35 950 207 149.472 7056 π F/m ≈ 8.854 187 8176 … × 10 − 12 F/m where c0 is the speed of light in free space, µ0 is the vacuum permeability. The constants c0 and μ0 are defined in SI units to have exact numerical values, shifting responsibility of experiment to the determination of the meter and the ampere; the linear permittivity of a homogeneous material is given relative to that of free space, as a relative permittivity εr (also called dielectric constant, although this term is deprecated and sometimes only refers to the static, zero-frequenc
Radiation pattern
In the field of antenna design the term radiation pattern refers to the directional dependence of the strength of the radio waves from the antenna or other source. In the fields of fiber optics and integrated optics, the term radiation pattern may be used as a synonym for the near-field pattern or Fresnel pattern; this refers to the positional dependence of the electromagnetic field in the near-field, or Fresnel region of the source. The near-field pattern is most defined over a plane placed in front of the source, or over a cylindrical or spherical surface enclosing it; the far-field pattern of an antenna may be determined experimentally at an antenna range, or alternatively, the near-field pattern may be found using a near-field scanner, the radiation pattern deduced from it by computation. The far-field radiation pattern can be calculated from the antenna shape by computer programs such as NEC. Other software, like HFSS can compute the near field; the far field radiation pattern may be represented graphically as a plot of one of a number of related variables, including.
Only the relative amplitude is plotted, normalized either to the amplitude on the antenna boresight, or to the total radiated power. The plotted quantity may be shown on a linear scale, or in dB; the plot is represented as a three-dimensional graph, or as separate graphs in the vertical plane and horizontal plane. This is known as a polar diagram, it is a fundamental property of antennas that the receiving pattern of an antenna when used for receiving is identical to the far-field radiation pattern of the antenna when used for transmitting. This is proved below. Therefore, in discussions of radiation patterns the antenna can be viewed as either transmitting or receiving, whichever is more convenient. Note however that this applies only to the passive antenna elements. Active antennas that include amplifiers or other components are no longer reciprocal devices. Since electromagnetic radiation is dipole radiation, it is not possible to build an antenna that radiates coherently in all directions, although such a hypothetical isotropic antenna is used as a reference to calculate antenna gain.
The simplest antennas and dipole antennas, consist of one or two straight metal rods along a common axis. These axially symmetric antennas have radiation patterns with a similar symmetry, called omnidirectional patterns; this illustrates the general principle that if the shape of an antenna is symmetrical, its radiation pattern will have the same symmetry. In most antennas, the radiation from the different parts of the antenna interferes at some angles; this results in zero radiation at certain angles where the radio waves from the different parts arrive out of phase, local maxima of radiation at other angles where the radio waves arrive in phase. Therefore, the radiation plot of most antennas shows a pattern of maxima called "lobes" at various angles, separated by "nulls" at which the radiation goes to zero; the larger the antenna is compared to a wavelength, the more lobes there will be. In a directive antenna in which the objective is to direct the radio waves in one particular direction, the lobe in that direction is larger than the others.
The axis of maximum radiation, passing through the center of the main lobe, is called the "beam axis" or boresight axis". In some antennas, such as split-beam antennas, there may exist more than one major lobe. A minor lobe is any lobe except a major lobe; the other lobes, representing unwanted radiation in other directions, are called "side lobes". The side lobe in the opposite direction from the main lobe is called the "back lobe". Minor lobes represent radiation in undesired directions, so in directional antennas a design goal is to reduce the minor lobes. Side lobes are the largest of the minor lobes; the level of minor lobes is expressed as a ratio of the power density in the lobe in question to that of the major lobe. This ratio is termed the side lobe ratio or side lobe level. Side lobe levels of −20 dB or greater are not desirable in many applications. Attainment of a side lobe level smaller than −30 dB requires careful design and construction. In most radar systems, for example, low side lobe ratios are important to minimize false target indications through the side lobes.
For a complete proof, see the reciprocity article. Here, we present a common simple proof limited to the approximation of two antennas separated by a large distance compared to the size of the antenna, in a homogeneous medium; the first antenna is the test antenna. The second antenna is a reference antenna; each antenna is alternately connected to a transmitter having a particular source impedance, a receiver having the same input impedance. It is assumed that the two antennas are sufficiently far apart that the properties of the transmitting antenna are not affected by the load placed upon it by the receiving antenna; the amount of power transferred from the transmitter to the receiver c
Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics and fluid dynamics; the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, Joseph-Louis Lagrange. In 1746, d’Alembert discovered the one-dimensional wave equation, within ten years Euler discovered the three-dimensional wave equation; the wave equation is a partial differential equation that may constrain some scalar function u = u of a time variable t and one or more spatial variables x1, x2, … xn. The quantity u may be, for example, the pressure in a liquid or gas, or the displacement, along some specific direction, of the particles of a vibrating solid away from their resting positions; the equation is ∂ 2 u ∂ t 2 = c 2. Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as u ¨ = c 2 ∇ 2 u where ◻ ¨ denotes double time derivative, ∇ is the nabla operator, ∇2 = ∇ · ∇ is the Laplacian operator: u ¨ = ∂ 2 u ∂ t 2 ∇ = ∇ 2 = ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 + ⋯ + ∂ 2 ∂ x n 2 A solution of this equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed c.
This analysis is possible. This property is called the superposition principle in physics; the wave equation alone does not specify a physical solution. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments; the wave equation is the simplest example of a hyperbolic differential equation. It, its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity and many other scientific and technical disciplines; the wave equation in one space dimension can be written as follows: ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2. This equation is described as having only one space dimension x, because the only other independent variable is the time t; the dependent variable u may represent a second space dimension, if, for example, the displacement u takes place in y-direction, as in the case of a string, located in the x–y plane.
The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string, vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension. Another physical setting for deriva
Electromagnetic radiation
In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space, carrying electromagnetic radiant energy. It includes radio waves, infrared, ultraviolet, X-rays, gamma rays. Classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light, which, in a vacuum, is denoted c. In homogeneous, isotropic media, the oscillations of the two fields are perpendicular to each other and perpendicular to the direction of energy and wave propagation, forming a transverse wave; the wavefront of electromagnetic waves emitted from a point source is a sphere. The position of an electromagnetic wave within the electromagnetic spectrum can be characterized by either its frequency of oscillation or its wavelength. Electromagnetic waves of different frequency are called by different names since they have different sources and effects on matter. In order of increasing frequency and decreasing wavelength these are: radio waves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.
Electromagnetic waves are emitted by electrically charged particles undergoing acceleration, these waves can subsequently interact with other charged particles, exerting force on them. EM waves carry energy and angular momentum away from their source particle and can impart those quantities to matter with which they interact. Electromagnetic radiation is associated with those EM waves that are free to propagate themselves without the continuing influence of the moving charges that produced them, because they have achieved sufficient distance from those charges. Thus, EMR is sometimes referred to as the far field. In this language, the near field refers to EM fields near the charges and current that directly produced them electromagnetic induction and electrostatic induction phenomena. In quantum mechanics, an alternate way of viewing EMR is that it consists of photons, uncharged elementary particles with zero rest mass which are the quanta of the electromagnetic force, responsible for all electromagnetic interactions.
Quantum electrodynamics is the theory of. Quantum effects provide additional sources of EMR, such as the transition of electrons to lower energy levels in an atom and black-body radiation; the energy of an individual photon is greater for photons of higher frequency. This relationship is given by Planck's equation E = hν, where E is the energy per photon, ν is the frequency of the photon, h is Planck's constant. A single gamma ray photon, for example, might carry ~100,000 times the energy of a single photon of visible light; the effects of EMR upon chemical compounds and biological organisms depend both upon the radiation's power and its frequency. EMR of visible or lower frequencies is called non-ionizing radiation, because its photons do not individually have enough energy to ionize atoms or molecules or break chemical bonds; the effects of these radiations on chemical systems and living tissue are caused by heating effects from the combined energy transfer of many photons. In contrast, high frequency ultraviolet, X-rays and gamma rays are called ionizing radiation, since individual photons of such high frequency have enough energy to ionize molecules or break chemical bonds.
These radiations have the ability to cause chemical reactions and damage living cells beyond that resulting from simple heating, can be a health hazard. James Clerk Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields and their symmetry; because the speed of EM waves predicted by the wave equation coincided with the measured speed of light, Maxwell concluded that light itself is an EM wave. Maxwell's equations were confirmed by Heinrich Hertz through experiments with radio waves. According to Maxwell's equations, a spatially varying electric field is always associated with a magnetic field that changes over time. A spatially varying magnetic field is associated with specific changes over time in the electric field. In an electromagnetic wave, the changes in the electric field are always accompanied by a wave in the magnetic field in one direction, vice versa; this relationship between the two occurs without either type of field causing the other.
In fact, magnetic fields can be viewed as electric fields in another frame of reference, electric fields can be viewed as magnetic fields in another frame of reference, but they have equal significance as physics is the same in all frames of reference, so the close relationship between space and time changes here is more than an analogy. Together, these fields form a propagating electromagnetic wave, which moves out into space and need never again interact with the source; the distant EM field formed in this way by the acceleration of a charge carries energy with it that "radiates" away through space, hence the term. Maxwell's equations established that some charges and currents produce a local type of electromagnetic field near them that does not have the behaviour of EMR. Currents directly produce a magnetic field, but it is of a magnetic dipole type that dies out with distance from the current. In a similar manner, moving charges pushed apart in a conductor by a changing electrical potential produce an electric dipole type electric
Frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency; the period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals, radio waves, light. For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics and radio, frequency is denoted by a Latin letter f or by the Greek letter ν or ν; the relation between the frequency and the period T of a repeating event or oscillation is given by f = 1 T.
The SI derived unit of frequency is the hertz, named after the German physicist Heinrich Hertz. One hertz means. If a TV has a refresh rate of 1 hertz the TV's screen will change its picture once a second. A previous name for this unit was cycles per second; the SI unit for period is the second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz. As a matter of convenience and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are described by their frequency instead of period; these used conversions are listed below: Angular frequency denoted by the Greek letter ω, is defined as the rate of change of angular displacement, θ, or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument to the sine function: y = sin = sin = sin d θ d t = ω = 2 π f Angular frequency is measured in radians per second but, for discrete-time signals, can be expressed as radians per sampling interval, a dimensionless quantity.
Angular frequency is larger than regular frequency by a factor of 2π. Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.: y = sin = sin d θ d x = k Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has an inverse relationship to the wavelength, λ. In dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ. In the special case of electromagnetic waves moving through a vacuum v = c, where c is the speed of light in a vacuum, this expression becomes: f = c λ; when waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can done in the following ways, Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period dividing the count by the length of the time period.
For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz If the number of counts is not large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count; this is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T
Speed of light
The speed of light in vacuum denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second, it is exact because by international agreement a metre is defined as the length of the path travelled by light in vacuum during a time interval of 1/299792458 second. According to special relativity, c is the maximum speed at which all conventional matter and hence all known forms of information in the universe can travel. Though this speed is most associated with light, it is in fact the speed at which all massless particles and changes of the associated fields travel in vacuum; such particles and waves travel at c regardless of the motion of the source or the inertial reference frame of the observer. In the special and general theories of relativity, c interrelates space and time, appears in the famous equation of mass–energy equivalence E = mc2; the speed at which light propagates through transparent materials, such as glass or air, is less than c.
The ratio between c and the speed v at which light travels in a material is called the refractive index n of the material. For example, for visible light the refractive index of glass is around 1.5, meaning that light in glass travels at c / 1.5 ≈ 200,000 km/s. For many practical purposes and other electromagnetic waves will appear to propagate instantaneously, but for long distances and sensitive measurements, their finite speed has noticeable effects. In communicating with distant space probes, it can take minutes to hours for a message to get from Earth to the spacecraft, or vice versa; the light seen from stars left them many years ago, allowing the study of the history of the universe by looking at distant objects. The finite speed of light limits the theoretical maximum speed of computers, since information must be sent within the computer from chip to chip; the speed of light can be used with time of flight measurements to measure large distances to high precision. Ole Rømer first demonstrated in 1676 that light travels at a finite speed by studying the apparent motion of Jupiter's moon Io.
In 1865, James Clerk Maxwell proposed that light was an electromagnetic wave, therefore travelled at the speed c appearing in his theory of electromagnetism. In 1905, Albert Einstein postulated that the speed of light c with respect to any inertial frame is a constant and is independent of the motion of the light source, he explored the consequences of that postulate by deriving the theory of relativity and in doing so showed that the parameter c had relevance outside of the context of light and electromagnetism. After centuries of precise measurements, in 1975 the speed of light was known to be 299792458 m/s with a measurement uncertainty of 4 parts per billion. In 1983, the metre was redefined in the International System of Units as the distance travelled by light in vacuum in 1/299792458 of a second; the speed of light in vacuum is denoted by a lowercase c, for "constant" or the Latin celeritas. In 1856, Wilhelm Eduard Weber and Rudolf Kohlrausch had used c for a different constant shown to equal √2 times the speed of light in vacuum.
The symbol V was used as an alternative symbol for the speed of light, introduced by James Clerk Maxwell in 1865. In 1894, Paul Drude redefined c with its modern meaning. Einstein used V in his original German-language papers on special relativity in 1905, but in 1907 he switched to c, which by had become the standard symbol for the speed of light. Sometimes c is used for the speed of waves in any material medium, c0 for the speed of light in vacuum; this subscripted notation, endorsed in official SI literature, has the same form as other related constants: namely, μ0 for the vacuum permeability or magnetic constant, ε0 for the vacuum permittivity or electric constant, Z0 for the impedance of free space. This article uses c for the speed of light in vacuum. Since 1983, the metre has been defined in the International System of Units as the distance light travels in vacuum in 1⁄299792458 of a second; this definition fixes the speed of light in vacuum at 299,792,458 m/s. As a dimensional physical constant, the numerical value of c is different for different unit systems.
In branches of physics in which c appears such as in relativity, it is common to use systems of natural units of measurement or the geometrized unit system where c = 1. Using these units, c does not appear explicitly because multiplication or division by 1 does not affect the result; the speed at which light waves propagate in vacuum is independent both of the motion of the wave source and of the inertial frame of reference of the observer. This invariance of the speed of light was postulated by Einstein in 1905, after being motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous aether, it is only possible to verify experimentally that the two-way speed of light is frame-independent, because it is impossible to measure the one-way speed of light without some convention as to how clocks at the source and at the detector should be synchronized. However