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Νέα σελίδα: Στη φυσική και στη μαθηματική ανάλυση, ο '''Νόμος του Γκάους''' είναι η εφαρμογή του γενικευμέν...
Γραμμή 1:
Στη [[φυσική]] και στη [[μαθηματική ανάλυση]], ο '''Νόμος του Γκάους''' είναι η εφαρμογή του γενικευμένου [[θεώρημα της απόκλισης|θεωρήματος της απόκλισης]] στην [[ηλεκτροστατική]], δίνοντας την σχέση ισοδυναμίας μεταξύ μιας οποιαδήποτε [[ροή|ροής]], όπως ενός [[υγρό|υγρού]], της ηλεκτρικής ή της βαρυτικής, που ρέει έξω από μια οποιαδήποτε κλειστή επιφάνεια και το αποτέλεσμα των εσωτερικών πηγών, όπως το [[ηλεκτρικό φορτίο]] ή η μάζα, που περιέχονται στον όγκο που περικλύει η επιφάνεια. Ο νόμος αναπτύχθηκε από τον [[Καρλ Φρίντριχ Γκάους]].
In [[physics]] and [[mathematical analysis]], '''Gauss's law''' gives the relation between the electric [[flux]] flowing out a closed surface and the [[electric charge]] enclosed in the surface.
 
==Ολοκληρωτική μορφή==
==Integral Form==
 
Στην ολοκληρωτική του μορφή, ο νόμος λέει:
In its integral form, the law states:
 
: <math>\Phi = \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}
= {1 \over \epsilon_ovarepsilon_o} \int_V \rho\ dV\mathrm{d}V = \frac{Q_A}{\epsilon_ovarepsilon_o}</math>
 
whereόπου <math>\Phi</math> isείναι theη [[electricηλεκτρική flux]]ροή, <math>\mathbf{E}</math> isείναι theτο [[electricηλεκτρικό fieldπεδίο]], <math>\mathrm{d}\mathbf{A}</math> isείναι theη areaαπειροστή ofπεριοχή aτης differentialκλειστής square on the closed surfaceεπιφάνειας ''S'' with an outward facing [[surface normal]] defining its direction, <math>Q_\mathrm{A}</math> isείναι theτο chargeφορτίο enclosedπου byπερικλύει theη surfaceεπιφάνεια, <math>\rho</math> isείναι theη chargeηλεκτρική densityπυκνότητα atσε aένα pointσημείο inτου όγκου <math>V</math>, <math>\epsilon_ovarepsilon_o</math> isείναι theη [[permittivityεπιδεκτικότητα]] ofτου freeκενού spaceχώρου andκαι <math>\oint_S</math> isείναι theτο integralολοκλήρωμα overπάνω theστην surfaceκλειστή επιφάνεια ''S'', που περικλύει enclosingτον volumeόγκο ''V''.
'''[[Image:Bold text]][[Media:#REDIRECT [[Example.ogg]]
----
--[[User:61.1.236.142|61.1.236.142]] 01:48, 28 June 2006 (UTC)<nowiki>tyrt<math>ytr</math></nowiki>]]'''
 
Για πληροφορίες και τη στρατηγική της εφαρμογής του νόμου του Γκάους, δείτε τις [[Γκαουσιανή επιφάνεια|Γκαουσιανές επιφάνειες]].
==Differential Form==
 
==Διαφορική μορφή==
In [[partial differential equation|differential form]], the equation becomes:
 
Σε διαφορική μορφή, η εξίσωση γίνεται:
:<math>\nabla \cdot \mathbf{D} = \rho</math> ▼
 
▲:<math>\mathbf{\nabla} \cdot \mathbf{D} = \rhorho_{\mathrm{free}}</math>
where <math>\nabla</math> is the [[del operator]], representing [[divergence]], '''D''' is the [[electric displacement field]] (in units of C/m²), and '''ρ''' is the ''free'' electric charge density (in units of C/m³), not including [[dipole]] charges [[bound charge|bound]] in a material. The differential form derives in part from Gauss's [[divergence theorem]].
 
όπου <math>\mathbf{\nabla}</math> είναι το [[ανάδελτα]], <math>D</math> είναι το [[electric displacement field]] (σε μονάδες C/m²), και <math>\rho_{\mathrm{free}}</math> είναι η ''ελεύθερη'' πυκνότητα ηλεκτρικών φορτίων (σε μονάδες C/m³), που δε συμπεριλαμβάνει τα δέσμια [[δίπολο|διπολικά]] φορτία σε ένα υλικό.
 
Για γραμμικά υλικά, η εξίσωση γίνεται:
And for linear materials, the equation becomes:
 
:<math>\mathbf{\nabla} \cdot \epsilonvarepsilon \mathbf{E} = \rhorho_{\mathrm{free}}</math>
 
whereόπου <math>\epsilonvarepsilon</math> isείναι theη electricalηλεκτρική [[permittivityεπιδεκτικότητα]].
 
==Νόμος του Κουλόμπ==
==Coulomb's Law==
In the special case of a spherical surface with a central charge, the [[electric field]] is perpendicular to the surface, with the same magnitude at all points of it, giving the simpler expression:
 
:<math>E=\frac{Q}{4\pi\epsilon_0rvarepsilon_0r^{2}}</math>
 
where ''E'' is the [[electric field]] strength at radius ''r'', ''Q'' is the enclosed charge, and ε<sub>0</sub> is the permitivity of free space. Thus the familiar [[inverse-square law]] dependence of the electric field in [[Coulomb's law]] follows from Gauss's law.
Γραμμή 39 ⟶ 37 :
It was formulated by [[Carl Friedrich Gauss]] in [[1835]], but was not published until [[1867]]. Because of the mathematical similarity, Gauss's law has application for other physical quantities governed by an [[inverse-square law]] such as [[gravitation]] or the [[intensity]] of [[radiation]]. See also [[divergence theorem]].
 
==Εφαρμογή στο Μαγνητισμό==
==Gravitational Analogue==
In the static case of a bar magnet or other situation where the generator of a magnetic field is at rest with respect to the observer, the integral form of Gauss's Law can be proven using a heuristic argument regarding the net flux proportionality to the number of field lines that enter and leave a Gaussian surface.
 
With such an argument it can be shown that in all static cases, the net magnetic flux is zero. As many field lines enter any Gaussian surface as leave a Gaussian surface, and so there is no "source" of the magnetic field to enclose.
Since both gravity and electromagnetism propagate relative to the squared distance between two objects, we can relate the two using Gauss's Law by examining their respective vector fields <math>\mathbf{G}</math> and <math>\mathbf{E}</math>, where
 
: <math>\mathbf{G}Phi_B = -G_\oint_S \mathbf{cB} \fraccdot \mathrm{m}{r^2d}\hatmathbf{rA} = 0</math>,
 
The differential form of this is one of Maxwell's Equations, which is a consequence of the fact that magnetic monopoles do not exist.
and
 
==Εφαρμογή στη Βαρύτητα==
: <math>\mathbf{E} = \frac{1}{4 \pi \epsilon_{0}} \frac{q}{r^2}\hat{r}</math>,▼
The gravitational form of Gauss's Law is largely a theoretical curiosity, but can be used by analogy to the electrostatic form of Gauss's Law to prove that the gravitational force of any body on any other body can be treated as though both masses were concentrated at their centers.
 
: <math>\Phi_{G}Phi_g = \oint_S G(r) \hatmathbf{rg} \cdot \hatmathrm{rd} r^\mathbf{2A} d\Omega</math>,▼
where <math>G_{c}</math> is the [[gravitational constant]], <math>m</math> is the mass of the point source, <math>r</math> is the radius (distance) between the point source and another object, <math>\epsilon_{0}</math> is the [[permittivity of free space]], and <math>q</math> is the charge of the electric point source.
= 4 \pi G \int_V \rho_m\ \mathrm{d}V = 4 \pi GM</math>
 
In applying the sameabove wayform thatof weGauss's evaluateLaw theto surface integralprove, for electromagnetismexample, to getthat the resultforce <math>\frac{q}{\epsilon_{0}}</math>,of wethe canEarth chooseacting aon properthe [[GaussianMoon Surface]] to find an answer for thedoes gravitationalnot flux.depend Foron a pointdetailed masstreatment centered atof the coordinateEarth's system origincomposition, one encloses the mostEarth logicalin choice fora ourspherical Gaussian surface, iswhose aarea sphere of radiusis <math>4 \pi r^{2}</math> centered at the origin.
 
Since the field lines of the Earth extend out equally in all directions and fall off as <math> \frac{1}{r^{2}} </math> (which can be proven independently from Newtonian mechanics and the force law so derived), the gravitational field must be constant at a given radius.
We start with the integral form of Gauss's Law
 
: <math>\Phi_{G} = \oint_S \mathbf{Gg} \cdot \mathrm{d}\mathbf{A} = 4 \pi GM </math>.
: <math> \mathbf{g} \oint_S \cdot \mathrm{d}\mathbf{A} = 4 \pi GM </math>
: <math>d \mathbf{Ag} = 4 \pi r^{2} d\Omega= 4 \hat{r}pi GM </math>.▼
▲: <math> \mathbf{Eg} = \frac{1}{4 \pi \epsilon_{0}} \frac{qGM}{r^{2}\hat{r} </math>,
 
Trivially, multiplying through by m yields the familiar force equation. If the only assumption being made is that gravitational field lines look like electrostatic ones then no prior knowledge of Newton's work is needed. While no reference can be formally found for this, it is often remarked casually in introductory physics classes that Isaac Newton took several pages of calculus to prove that mass distributions act as though their mass were concentrated at a point in their center as far as their interactions with other bodies are concerned, and that had he had Gauss's Law, much of the cumbersome work he undertook would have been shortened dramatically.
An infinitesimal area element is just the area of the infinitesimal solid angle, which is defined as
 
==Δείτε επίσης==
▲: <math>d\mathbf{A} = r^{2} d\Omega \hat{r}</math>.
* [[Εξισώσεις Μάξγουελ]]
* [[Γκαουσιανή επιφάνεια]]
* [[Καρλ Φρίντριχ Γκάους]]
* [[Θεώρημα της απόκλισης]]
* [[FluxΡοή]]▼
* [[Μέθοδος των ειδώλων]]
 
==Εξωτερικοί σύνδεσμοι==
Our Gaussian Surface is wisely chosen since the vector normal to the surface is radial from the origin. With
* [http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoLectures/index.htm MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism] Taught by Professor [[Walter Lewin]].
 
*[http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html#Section10.6 section on Gauss's law in an online textbook]
▲: <math>\Phi_{G} = \oint_S G(r) \hat{r} \cdot \hat{r} r^{2} d\Omega</math>,
 
we see the inner product of the two radial vectors is unity and that both the magnitude of our field, <math>\mathbf{G}</math>, and the square of the distance between the surface and the point, <math>r^{2}</math>, remain constant over every element of the surface. This gives us the integral
 
: <math>\Phi_{G} = G(r) r^{2} \oint_S d\Omega</math>.
 
The remaining surface integral is just the surface area of our sphere (<math>4 \pi r^{2}</math>). If we combine this with our gravitational field equation from above, we have an expression for the gravitational flux of a point mass.
 
: <math>\Phi_{G} = -\frac{G_{c}m}{r^2} 4 \pi r^{2} = -4\pi G_{c}m</math>
 
It is interesting to note that the gravitational flux, like its electromagnetic counterpart, does not depend on the radius of the sphere.
 
==See also==
* [[Maxwell's equations]]
* [[Gaussian surface]]
* [[Carl Friedrich Gauss]]
* [[Divergence theorem]]
▲* [[Flux]]
* [[Method of image charges]]
 
==External links==
*[http://35.9.69.219/home/modules/pdf_modules/m132.pdf <small>MISN-0-132</small> ''Gauss's Law for Spherical Symmetry''] ([[Portable Document Format|PDF file]]) by Peter Signell for [http://www.physnet.org Project PHYSNET].
*[http://35.9.69.219/home/modules/pdf_modules/m133.pdf <small>MISN-0-133</small> ''Gauss's Law Applied to Cylindrical and Planar Charge Distributions] (PDF file) by Peter Signell for Project PHYSNET.
 
[[Κατηγορία:Ηλεκτροστατική]]
[[Category:Electrostatics]]
[[Category:Vector calculus]]
[[Category:Eponymous laws]]
[[Category:Introductory physics]]
 
[[ar:قانون غاوس]]
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[[fr:Théorème de Gauss (électromagnétisme)]]
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Γραμμή 101 ⟶ 92 :
[[it:Teorema di Gauss]]
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[[ja:ガウスの法則]]
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Ανακτήθηκε από "https://el.wikipedia.org/wiki/Νόμος_του_Γκάους"