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63-745: It is mentioned in the 1513 treatise Obra de Agricultura by Gabriel Alonso de Herrera . It may have originated in the Alentejo region of Portugal, although it is now rare in that area. Cayetana Blanca was the third most planted white grape variety in Spain, with 39,919 ha (98,640 acres) in 2015, totalling 4% of the grapes and 9% of the white variety hectarage . Cayetana blanca is also known under several synonyms, including multiple spelling variant for each: Other synonyms include Aujubi, Dedo or Dedro, Hoja vuelta, Mariouti, Neruca and Tierra de Barros. However, some synonyms can lead to confusion. In Spain, Cayetana blanca

126-538: 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 (the first two ones explicitly only in the SI formulation) are: In the differential equations, In the integral equations, The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over

189-490: A changing magnetic field and generates an electric field in a nearby wire. The original law of Ampère states that magnetic fields relate to electric current . Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current . The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve. Maxwell's modification of Ampère's circuital law

252-406: A changing magnetic field through Maxwell's modification of Ampère's circuital law . This perpetual cycle allows these waves, now known as electromagnetic radiation , to move through space at velocity c . The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This

315-1288: A constant speed in vacuum, c ( 299 792 458  m/s ). Known as electromagnetic radiation , these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays . In partial differential equation form and a coherent system of units , Maxwell's microscopic equations can be written as ∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 ( J + ε 0 ∂ E ∂ t ) {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}} With E {\displaystyle \mathbf {E} }

378-871: A factor (see Heaviside–Lorentz units , used mainly in particle physics ). The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem . According to the (purely mathematical) Gauss divergence theorem , the electric flux through the boundary surface ∂Ω can be rewritten as The integral version of Gauss's equation can thus be rewritten as ∭ Ω ( ∇ ⋅ E − ρ ε 0 ) d V = 0 {\displaystyle \iiint _{\Omega }\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\,\mathrm {d} V=0} Since Ω

441-869: A fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume. The definitions of the auxiliary fields are: D ( r , t ) = ε 0 E ( r , t ) + P ( r , t ) , H ( r , t ) = 1 μ 0 B ( r , t ) − M ( r , t ) , {\displaystyle {\begin{aligned}\mathbf {D} (\mathbf {r} ,t)&=\varepsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t),\\\mathbf {H} (\mathbf {r} ,t)&={\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t),\end{aligned}}} where P

504-767: A given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law: d d t ∬ Σ B ⋅ d S = ∬ Σ ∂ B ∂ t ⋅ d S , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,,} Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using

567-401: A time-varying magnetic field corresponds to curl of an electric field . In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface. The electromagnetic induction is the operating principle behind many electric generators : for example, a rotating bar magnet creates

630-544: Is also known as Jaén (including the variants Jaén blanco and Jaén rosado), but his is ambiguous since there are other varieties with the same name, such as Mencía (from El Bierzo and Ribeira Sacra ), which is known as Jaén colorado in Léon and as Jaen do Dão in Dão , Portugal, and Jaén tinto from Andalusia. In English sources Jaén with the Spanish accent often refers to Cayetana, while

693-553: Is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement. Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives which is satisfied for all Ω if and only if ∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0} everywhere. By

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756-443: Is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space . The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches

819-443: Is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping. The microscopic version is sometimes called "Maxwell's equations in vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive

882-432: Is that the additional fields D and H need to be determined through phenomenological constituent equations relating these fields to the electric field E and the magnetic field B , together with the bound charge and current. See below for a detailed description of the differences between the microscopic equations, dealing with total charge and current including material contributions, useful in air/vacuum; and

945-807: Is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρ b and bound current density J b in terms of polarization P and magnetization M are then defined as ρ b = − ∇ ⋅ P , J b = ∇ × M + ∂ P ∂ t . {\displaystyle {\begin{aligned}\rho _{\text{b}}&=-\nabla \cdot \mathbf {P} ,\\\mathbf {J} _{\text{b}}&=\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}.\end{aligned}}} If we define

1008-447: Is usually less than c . In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law . In turn, that electric field creates

1071-801: The Ampère–Maxwell law , the modified version of Ampère's circuital law, in integral form can be rewritten as ∬ Σ ( ∇ × B − μ 0 ( J + ε 0 ∂ E ∂ t ) ) ⋅ d S = 0. {\displaystyle \iint _{\Sigma }\left(\nabla \times \mathbf {B} -\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)\cdot \mathrm {d} \mathbf {S} =0.} Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that

1134-629: The Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls ) over a surface it bounds, i.e. ∮ ∂ Σ B ⋅ d ℓ = ∬ Σ ( ∇ × B ) ⋅ d S , {\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\iint _{\Sigma }(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} ,} Hence

1197-573: The Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside , has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x , y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For

1260-419: The Lorentz force law, form the foundation of classical electromagnetism , classical optics , electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges , currents , and changes of

1323-446: The magnetization M . The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and M , which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, Maxwell's macroscopic equations ignore many details on

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1386-629: The old SI system of units, the values of μ 0 = 4 π × 10 − 7 {\displaystyle \mu _{0}=4\pi \times 10^{-7}} and c = 299 792 458   m/s {\displaystyle c=299\,792\,458~{\text{m/s}}} are defined constants, (which means that by definition ε 0 = 8.854 187 8... × 10 − 12   F/m {\displaystyle \varepsilon _{0}=8.854\,187\,8...\times 10^{-12}~{\text{F/m}}} ) that define

1449-448: 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 theories of electromagnetism and optics . In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature ,

1512-481: The vacuum permeability . The equations have two major variants: The term "Maxwell's equations" is often also used for equivalent alternative formulations . Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem , analytical mechanics , or for use in quantum mechanics . The covariant formulation (on spacetime rather than space and time separately) makes

1575-1085: The Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: In particular, in an isolated system the total charge is conserved. In a region with no charges ( ρ = 0 ) and no currents ( J = 0 ), such as in vacuum, Maxwell's equations reduce to: ∇ ⋅ E = 0 , ∇ × E + ∂ B ∂ t = 0 , ∇ ⋅ B = 0 , ∇ × B − μ 0 ε 0 ∂ E ∂ t = 0. {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0,&\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0,\\\nabla \cdot \mathbf {B} &=0,&\nabla \times \mathbf {B} -\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=0.\end{aligned}}} Taking

1638-601: The Gaussian ( CGS ) units. Using these definitions, colloquially "in Gaussian units", the Maxwell equations become: The equations simplify slightly when a system of quantities is chosen in the speed of light, c , is used for nondimensionalization , so that, for example, seconds and lightseconds are interchangeable, and c = 1. Further changes are possible by absorbing factors of 4 π . This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such

1701-461: The Latin word tractatus , which is a form of the verb tractare , meaning "to handle," "to manage," or "to deal with". The Latin roots suggest a connotation of engaging with or discussing a subject in depth, which aligns with the modern understanding of a treatise as a formal and systematic written discourse on a specific topic. The works presented here have been identified as influential by scholars on

1764-475: The Portuguese spelling without the accent ( Jaen ) refers to Mencía. Avesso , which may be related to Jaen, is also often mistaken for Cayetana blanca. Albillo Mayor ( Ribera del Duero ), Calagraño (La Rioja), Doradilla and Xarel·lo from Catalonia are commonly confused as well. Treatise A treatise is a formal and systematic written discourse on some subject concerned with investigating or exposing

1827-556: The ampere and the metre. In the new SI system, only c keeps its defined value, and the electron charge gets a defined value. In materials with relative permittivity , ε r , and relative permeability , μ r , the phase velocity of light becomes v p = 1 μ 0 μ r ε 0 ε r , {\displaystyle v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}},} which

1890-422: The atoms, most notably their electrons . The connection to angular momentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These bound currents can be described using

1953-471: The compatibility of Maxwell's equations with special relativity manifest . Maxwell's equations in curved spacetime , commonly used in high-energy and gravitational physics , are compatible with general relativity . In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences. The publication of

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2016-914: The curl (∇×) of the curl equations, and using the curl of the curl identity we obtain μ 0 ε 0 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , μ 0 ε 0 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}} The quantity μ 0 ε 0 {\displaystyle \mu _{0}\varepsilon _{0}} has

2079-401: The defining relations above to eliminate D , and H , the "macroscopic" Maxwell's equations reproduce the "microscopic" equations. In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and the electric field E , as well as the magnetizing field H and the magnetic field B . Equivalently, we have to specify

2142-422: The dependence of the polarization P (hence the bound charge) and the magnetization M (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations . For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for

2205-817: The development of human civilization. Euclid's Elements has appeared in more editions than any other books except the Bible and is one of the most important mathematical treatises ever. It has been translated to numerous languages and remains continuously in print since the beginning of printing. Before the invention of the printing press, it was manually copied and widely circulated. When scholars recognized its excellence, they removed inferior works from circulation in its favor. Many subsequent authors, such as Theon of Alexandria , made their own editions, with alterations, comments, and new theorems or lemmas. Many mathematicians were influenced and inspired by Euclid's masterpiece. For example, Archimedes of Syracuse and Apollonius of Perga ,

2268-399: The differential version and using Gauss and Stokes formula appropriately. The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε 0 and μ 0 into the units (and thus redefining these). With a corresponding change in the values of the quantities for the Lorentz force law this yields

2331-960: The dimension (T/L) . Defining c = ( μ 0 ε 0 ) − 1 / 2 {\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}} , the equations above have the form of the standard wave equations 1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , 1 c 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}} Already during Maxwell's lifetime, it

2394-457: The electric field, B {\displaystyle \mathbf {B} } the magnetic field, ρ {\displaystyle \rho } the electric charge density and J {\displaystyle \mathbf {J} } the current density . ε 0 {\displaystyle \varepsilon _{0}} is the vacuum permittivity and μ 0 {\displaystyle \mu _{0}}

2457-1237: The equations depend only on the free charges Q f and free currents I f . This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J ) into free and bound parts: Q = Q f + Q b = ∭ Ω ( ρ f + ρ b ) d V = ∭ Ω ρ d V , I = I f + I b = ∬ Σ ( J f + J b ) ⋅ d S = ∬ Σ J ⋅ d S . {\displaystyle {\begin{aligned}Q&=Q_{\text{f}}+Q_{\text{b}}=\iiint _{\Omega }\left(\rho _{\text{f}}+\rho _{\text{b}}\right)\,\mathrm {d} V=\iiint _{\Omega }\rho \,\mathrm {d} V,\\I&=I_{\text{f}}+I_{\text{b}}=\iint _{\Sigma }\left(\mathbf {J} _{\text{f}}+\mathbf {J} _{\text{b}}\right)\cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} .\end{aligned}}} The cost of this splitting

2520-412: The equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics . Gauss's law describes

2583-556: The fields. The equations are named after the physicist and mathematician James Clerk Maxwell , who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside . Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at

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2646-1625: The fluid is the curl of the velocity field. The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by the div–curl identity . Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: 0 = ∇ ⋅ ( ∇ × B ) = ∇ ⋅ ( μ 0 ( J + ε 0 ∂ E ∂ t ) ) = μ 0 ( ∇ ⋅ J + ε 0 ∂ ∂ t ∇ ⋅ E ) = μ 0 ( ∇ ⋅ J + ∂ ρ ∂ t ) {\displaystyle 0=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}\right)} i.e., ∂ ρ ∂ t + ∇ ⋅ J = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.} By

2709-513: The geometry of three-dimensional objects such as polyhedra), number theory, and the theory of proportions. It was essentially a compilation of all mathematics known to the Greeks up until Euclid's time. Drawing on the work of his predecessors, especially the experimental research of Michael Faraday , the analogy with heat flow by William Thomson (later Lord Kelvin) and the mathematical analysis of George Green , James Clerk Maxwell synthesized all that

2772-427: The greatest mathematicians of their time, received their training from Euclid's students and his Elements and were able to solve many open problems at the time of Euclid. It is a prime example of how to write a text in pure mathematics, featuring simple and logical axioms, precise definitions, clearly stated theorems, and logical deductive proofs. The Elements consists of thirteen books dealing with geometry (including

2835-413: The integrand is zero if and only if the Ampère–Maxwell law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical fluid dynamics : the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of

2898-424: The macroscopic equations, dealing with free charge and current, practical to use within materials. When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction. This produces a macroscopic bound charge in

2961-416: The macroscopic properties of bulk matter from its microscopic constituents. "Maxwell's macroscopic equations", also known as Maxwell's equations in matter , are more similar to those that Maxwell introduced himself. In the macroscopic equations, the influence of bound charge Q b and bound current I b is incorporated into the displacement field D and the magnetizing field H , while

3024-463: The magnetic field of a material is attributed to a dipole , and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field . The Maxwell–Faraday version of Faraday's law of induction describes how

3087-425: The material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive bound charge on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the polarization P of

3150-409: The material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P , a charge is also produced in the bulk. Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the components of

3213-409: 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 . Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated. The equations introduce the electric field , E ,

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3276-461: The phenomenon of light dispersion where other models failed. John William Strutt (Lord Rayleigh) and Josiah Willard Gibbs then proved that the optical equations derived from Maxwell's theory are the only self-consistent description of the reflection, refraction, and dispersion of light consistent with experimental results. Optics thus found a new foundation in electromagnetism . Hertz's experimental work in electromagnetism stimulated interest in

3339-513: The possibility of wireless communication, which did not require long and expensive cables and was faster than even the telegraph. Guglielmo Marconi adapted Hertz's equipment for this purpose in the 1890s. He achieved the first international wireless transmission between England and France in 1900 and by the following year, he succeeded in sending messages in Morse code across the Atlantic. Seeing its value,

3402-585: The principles of the subject and its conclusions. A monograph is a treatise on a specialized topic. The word "treatise" has its origins in the early 14th century, derived from the Anglo-French term tretiz , which itself comes from the Old French traitis , meaning "treatise" or "account." This Old French term is rooted in the verb traitier , which means "to deal with" or "to set forth in speech or writing". The etymological lineage can be traced further back to

3465-567: The relationship between an electric field and electric charges : an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space . Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles ; no north or south magnetic poles exist in isolation. Instead,

3528-420: The same equations expressed using tensor calculus or differential forms (see § Alternative formulations ). The differential and integral formulations are mathematically equivalent; both are 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

3591-499: The same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor : the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with

3654-473: The shipping industry adopted this technology at once. Radio broadcasting became extremely popular in the twentieth century and remains in common use in the early twenty-first. But it was Oliver Heaviside , an enthusiastic supporter of Maxwell's electromagnetic theory, who deserves most of the credit for shaping how people understood and applied Maxwell's work for decades to come; he was responsible for considerable progress in electrical telegraphy, telephony, and

3717-441: The study of the propagation of electromagnetic waves. Independent of Gibbs, Heaviside assembled a set of mathematical tools known as vector calculus to replace the quaternions , which were in vogue at the time but which Heaviside dismissed as "antiphysical and unnatural." Maxwell%27s equations Maxwell's equations , or Maxwell–Heaviside equations , are a set of coupled partial differential equations that, together with

3780-434: The total, bound, and free charge and current density by ρ = ρ b + ρ f , J = J b + J f , {\displaystyle {\begin{aligned}\rho &=\rho _{\text{b}}+\rho _{\text{f}},\\\mathbf {J} &=\mathbf {J} _{\text{b}}+\mathbf {J} _{\text{f}},\end{aligned}}} and use

3843-423: Was confirmed by Heinrich Hertz . In the process, Hertz generated and detected what are now called radio waves and built crude radio antennas and the predecessors of satellite dishes. Hendrik Lorentz derived, using suitable boundary conditions, Fresnel's equations for the reflection and transmission of light in different media from Maxwell's equations. He also showed that Maxwell's theory succeeded in illuminating

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3906-539: Was found that the known values for ε 0 {\displaystyle \varepsilon _{0}} and μ 0 {\displaystyle \mu _{0}} give c ≈ 2.998 × 10 8   m/s {\displaystyle c\approx 2.998\times 10^{8}~{\text{m/s}}} , then already known to be the speed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In

3969-553: Was known about electricity and magnetism into a single mathematical framework, Maxwell's equations . Originally, there were 20 equations in total. In his Treatise on Electricity and Magnetism (1873), Maxwell reduced them to eight. Maxwell used his equations to predict the existence of electromagnetic waves, which travel at the speed of light. In other words, light is but one kind of electromagnetic wave. Maxwell's theory predicted there ought to be other types, with different frequencies. After some ingenious experiments, Maxwell's prediction

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