გაუსის კანონი: განსხვავება გადახედვებს შორის

არ არის რედაქტირების რეზიუმე
(ახალი გვერდი: ფიზიკაში, '''გაუსის კანონი''', რომელიც ასევე ცნობილია როგორც '''...)
 
<center>''ელექტრული ნაკადი ნებისმიერ ჩაკეტილ ზედაპირზე პროპორციული ამ ზედაპირის შიგნით არსებული [[მუხტი]]სა.''</center>
 
კანონი ფორმულირებული იქნა [[კარლ ფრიდრიხ გაუსი]]ს მიერ 1835 წელს, მაგრამ გამოქვეყნებული იქნა მხოლოდ 1867 წელს.<ref>{{cite book | author=Bellone, Enrico | title=A World on Paper: Studies on the Second Scientific Revolution | year=1980}}</ref> გაუსის კანონი არის [[მაქსველის განტოლებები|მაქსველის ოთხი განტოლებიდან]] ერთ-ერთი, რომლებიც შეადგენენ კლასიკური [[ელექტროდინამიკა|ელექტროდინამიკის]] საფუძველს. gausisგაუსის kanonisკანონის meSveobiTმეშვეობით aseveშესაძლებელია SeiZlebaგამოყვანილი gamoyvanili iqnasიქნას [[kulonisკულონის kanoniკანონი]],<ref>{{cite book|last1=Halliday|first1=David|last2=Resnick|first2=Robert|title=Fundamentals of Physics|publisher=John Wiley & Sons, Inc|year=1970|page=452–53}}</ref> და პირიქით.
 
გაუსის კანონს ინტეგრალური ფორმით აქვს სახე:
სადაც '''∇'''&thinsp;·&thinsp;'''E''' არის [[ელექტრული ველის დაძაბულობა|ელექტრული ველის დაძაბულობის]] [[დივერგენცია]], ხოლო ''ρ'' არის [[მუხტის სიმკვრივე]].
 
გაუსის კანონის ინტეგრალური და დიფერენციალური ფორმების ექვივალენტურობას განაპირობებს თეორემა ''დივერგენციის შესახებ'', რომელსაც ასევე ''გაუსის თეორემას'' უწოდებენ.
The integral and differential forms are related by the [[divergence theorem]], also called Gauss's theorem. Each of these forms can also be expressed two ways: In terms of a relation between the electric field '''E''' and the total electric charge, or in terms of the [[electric displacement field]] '''D''' and the [[free charge|''free'' electric charge]].
 
Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as [[Gauss's law for magnetism]] and [[Gauss's law for gravity]]. In fact, any "[[inverse-square law]]" can be formulated in a way similar to Gauss's law: For example, Gauss's law itself is essentially equivalent to the inverse-square [[Coulomb's law]], and Gauss's law for gravity is essentially equivalent to the inverse-square [[Newton's law of gravity]].
 
Gauss's law can be used to demonstrate that there is no electric field inside a [[Faraday cage]] with no electric charges. Gauss's law is something of an electrical analogue of [[Ampère's law]], which deals with magnetism.
 
==In terms of total charge==
 
===Integral form===
For a volume ''V'' with surface ''S'', Gauss's law states that
:<math>\Phi_{E,S} = \frac{Q}{\varepsilon_0}</math>
where Φ<sub>''E'',''S''</sub> is the [[electric flux]] through ''S'', ''Q'' is total charge inside ''V'', and ''ε''<sub>0</sub> is the [[electric constant]]. The electric flux is given by a [[surface integral]] over ''S'':
:<math>\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}</math>
where '''E''' is the electric field, d'''A''' is a vector representing an [[infinitesimal]] element of [[area]],{{#tag:ref|More specifically, the infinitesimal area is thought of as [[Plane (mathematics)|planar]] and with area d''A''. The vector d'''A''' is [[normal]] to this area element and has [[magnitude]] d''A''.<ref>{{cite book|last=Matthews|first=Paul|title=Vector Calculus|publisher=Springer|year=1998|isbn=3540761802}}</ref>|group="note"}} and · represents the [[dot product]].
 
====Applying the integral form====
{{main|Gaussian surface}}
{{seealso|Capacitance#Gauss's law}}
 
If the electric field is known everywhere, Gauss's law makes it quite easy, in principle, to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux.
 
However, much more often, it is the reverse problem that needs to be solved: The electric charge distribution is known, and the electric field needs to be computed. This is much more difficult, since if you know the total flux through a given surface, that gives almost no information about the electric field, which (for all you know) could go in and out of the surface in arbitrarily complicated patterns.
 
An exception is if there is some [[symmetry]] in the situation, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include cylindrical symmetry, planar symmetry, and spherical symmetry. See the article [[Gaussian surface]] for examples where these symmetries are exploited to compute electric fields.
 
===Differential form===
 
In [[partial differential equation|differential form]], Gauss's law states:
 
:<math>\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} </math>
 
where '''∇'''&thinsp;· denotes [[divergence]], '''E''' is the electric field, and ''ρ'' is the total electric [[charge density]] (including both free and bound charge), and ''ε''<sub>0</sub> is the electric constant. This is mathematically equivalent to the integral form, because of the [[divergence theorem]].
 
===Equivalence of integral and differential forms===
{{main|Divergence theorem}}
 
The integral and differential forms are mathematically equivalent, by the [[divergence theorem]]. Here is the argument more specifically:
 
The integral form of Gauss's law is:
:<math>\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{Q}{\varepsilon_0}</math>
for any closed surface ''S'' containing charge ''Q''. By the divergence theorem, this equation is equivalent to:
:<math>\iiint\limits_V \nabla \cdot \mathbf{E} \ \mathrm{d}V = \frac{Q}{\varepsilon_0}</math>
for any volume ''V'' containing charge ''Q''. By the relation between charge and charge density, this equation is equivalent to:
:<math>\iiint\limits_V \nabla \cdot \mathbf{E} \ \mathrm{d}V = \iiint\limits_V \frac{\rho}{\varepsilon_0} \ \mathrm{d}V</math>
for any volume ''V''. In order for this equation to be ''simultaneously true'' for ''every'' possible volume ''V'', it is necessary (and sufficient) for the integrands to be equal everywhere. Therefore, this equation is equivalent to:
:<math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}.</math>
Thus the integral and differential forms are equivalent.
 
==In terms of free charge==
===Free versus bound charge===
{{main|Electric polarization}}
 
The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in [[static electricity]], or the charge on a [[capacitor]] plate. In contrast, "bound charge" arises only in the context of [[dielectric]] (polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge".
 
Although microscopically, all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more "fundamental" Gauss's law, in terms of '''E''', is sometimes put into the equivalent form below, which is in terms of '''D''' and the free charge only.
 
===Integral form===
 
This formulation of Gauss's law states that, for any volume ''V'' in space, with surface ''S'', the following equation holds:
:<math>\Phi_{D,S} = Q_{\mathrm{free}},\!</math>
where Φ<sub>''D'',''S''</sub> is the flux of the [[electric displacement field]] '''D''' through ''S'', and ''Q''<sub>free</sub> is the free charge contained in ''V''.
The flux Φ<sub>''D'',''S''</sub> is defined analogously to the flux Φ<sub>''E'',''S''</sub> of the electric field '''E''' through ''S''. Specifically, it is given by the surface integral
:<math>\Phi_{D,S} = \oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A}.</math>
 
===Differential form===
 
The differential form of Gauss's law, involving free charge only, states:
:<math>\mathbf{\nabla} \cdot \mathbf{D} = \rho_{\mathrm{free}}</math>
 
where '''∇''' · '''D''' is the [[divergence]] of the electric displacement field, and ''ρ''<sub>free</sub> is the free electric charge density.
 
The differential form and integral form are mathematically equivalent. The proof primarily involves the [[divergence theorem]].
 
==Equivalence of total and free charge statements==
 
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge.
|-
|In this proof, we will show that the equation
:<math>\nabla\cdot \mathbf{E} = \rho/\epsilon_0</math>
is equivalent to the equation
:<math>\nabla\cdot\mathbf{D} = \rho_{\mathrm{free}}</math>
Note that we're only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem.
 
We introduce the [[polarization density]] '''P''', which has the following relation to '''E''' and '''D''':
:<math>\mathbf{D}=\epsilon_0 \mathbf{E} + \mathbf{P}</math>
and the following relation to the bound charge:
:<math>\rho_{\mathrm{bound}} = -\nabla\cdot \mathbf{P}</math>
Now, consider the three equations:
:<math>\rho_{\mathrm{bound}} = \nabla\cdot (-\mathbf{P})</math>
:<math>\rho_{\mathrm{free}} = \nabla\cdot \mathbf{D}</math>
:<math>\rho = \nabla \cdot(\epsilon_0\mathbf{E})</math>
The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true [[if and only if]] the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.
|}
 
===In linear materials===
 
In homogeneous, isotropic, nondispersive, [[linear material]]s, there is a nice, simple relationship between '''E''' and '''D''':
:<math>\varepsilon \mathbf{E} = \mathbf{D}</math>
where ''ε'' is the [[permittivity]] of the material. Under these circumstances, there is yet another pair of equivalent formulations of Gauss's law:
:<math>\Phi_{E,S} = \frac{Q_{\mathrm{free}}}{\varepsilon}</math>
:<math>\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho_{\mathrm{free}}}{\varepsilon}</math>
 
==Relation to Coulomb's law==
===Deriving Gauss's law from Coulomb's law===
 
Gauss's law can be derived from [[Coulomb's law]], which states that the electric field due to a stationary [[point charge]] is:
 
:<math>\mathbf{E}(\mathbf{r}) = \frac{q}{4\pi \epsilon_0} \frac{\mathbf{e_r}}{r^2}</math>
where
:'''e<sub>r</sub>''' is the radial [[unit vector]],
:''r'' is the radius, |'''r'''|,
:<math>\epsilon_0</math> is the [[electric constant]],
:''q'' is the charge of the particle, which is assumed to be located at the [[origin (mathematics)|origin]].
 
Using the expression from Coulomb's law, we get the total field at '''r''' by using an integral to sum the field at '''r''' due to the infinitesimal charge at each other point '''s''' in space, to give
 
:<math>\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} d^3 \mathbf{s}</math>
 
where <math>\rho</math> is the charge density. If we take the divergence of both sides of this equation with respect to '''r''', and use the known theorem<ref>See, for example, {{cite book | author=Griffiths, David J. | title=Introduction to Electrodynamics (3rd ed.) | publisher=Prentice Hall | year=1998 | id=ISBN 0-13-805326-X | page=50}}</ref>
 
:<math>\nabla \cdot \left(\frac{\mathbf{s}}{|\mathbf{s}|^3}\right) = 4\pi \delta(\mathbf{s})</math>
where δ('''s''') is the [[Dirac delta function]], the result is
 
:<math>\nabla\cdot\mathbf{E}(\mathbf{r}) = \frac{1}{\epsilon_0} \int \rho(\mathbf{s})\ \delta(\mathbf{r}-\mathbf{s})\ d^3 \mathbf{s}</math>
 
Using the "[[Dirac delta function#Delta function of more complicated arguments|sifting property]]" of the Dirac delta function, we arrive at
 
:<math>\nabla\cdot\mathbf{E}(\mathbf{r}) = \rho(\mathbf{r})/\epsilon_0</math>
 
which is the differential form of Gauss's law, as desired.
 
Note that since Coulomb's law only applies to ''stationary'' charges, there is no reason to expect Gauss's law to hold for moving charges ''based on this derivation alone''. In fact, Gauss's law ''does'' hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.
 
===Deriving Coulomb's law from Gauss's law===
 
Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the [[Curl (mathematics)|curl]] of '''E''' (see [[Helmholtz decomposition]] and [[Faraday's law of induction|Faraday's law]]). However, Coulomb's law ''can'' be proven from Gauss's law if it is assumed, in addition, that the electric field from a [[point charge]] is spherically-symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).
 
Taking ''S'' in the integral form of Gauss's law to be a spherical surface of radius ''r'', centered at the point charge ''Q'', we have
: <math>\oint_{S}\mathbf{E}\cdot d\mathbf{A} = Q/\varepsilon_0</math>
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
: <math>4\pi r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) = Q/\varepsilon_0</math>
where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] pointing radially away from the charge. Again by spherical symmetry, '''E''' points in the radial direction, and so we get
: <math>\mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi \varepsilon_0}\frac{\hat{\mathbf{r}}}{r^2}</math>
which is essentially equivalent to Coulomb's law. Thus the [[inverse-square law]] dependence of the electric field in Coulomb's law follows from Gauss's law.
 
==See also==
* [[Method of image charges]]
 
==Notes==
{{reflist|group=note}}
 
==სქოლიო==
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