კულონის კანონი: განსხვავება გადახედვებს შორის

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ხაზი 22:
სადაც ''r'' არის მანძილი მუხტებს შორის, ხოლო ''k''<sub>e</sub> არის პროპორციულობის კოეფიციენტი.
 
[[SI სისტემა]]ში პროპორციულობის კოეფიციენტი ''k''<sub>e</sub>, რომელსაც ''კულონის მუდმივა'' ქვია მოიცემა შემდეგი ფორმულით
A positive force implies a repulsive interaction, while a negative force implies an attractive interaction.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c1 Coulomb's law], Hyperphysics</ref>
 
The proportionality constant ''k''<sub>e</sub>, called the '''Coulomb constant''' (sometimes called the '''Coulomb force constant'''), is related to defined [[Free space#Properties of free space|properties of space]] and can be calculated exactly:<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c3 Coulomb's constant], Hyperphysics</ref>
 
:<math> \begin{align}
Line 30 ⟶ 28:
&= 8.987\ 551\ 787\ 368\ 176\ 4 \times 10^9 \ \mathrm{N \cdot m^2 \cdot C^{-2}}. \\
\end{align} </math>
 
By definition in [[SI]] units, the [[speed of light in vacuum]], denoted ''c'',<ref name=note> Current practice is to use ''c''<sub>0</sub> to denote the speed of light in vacuum according to [[ISO 31]]. In the original Recommendation of 1983, the symbol ''c'' was used for this purpose and continues to be commonly used. See [http://physics.nist.gov/Pubs/SP330/sp330.pdf NIST ''Special Publication 330'', Appendix 2, p. 45 ]</ref> is {{val|299792458|u=[[meter|m]]·[[second|s]]<sup>−1</sup>}},<ref>[http://physics.nist.gov/cuu/Units/meter.html]</ref> and the [[magnetic constant]] (''μ''<sub>0</sub>), is defined as {{nowrap|4π × 10<sup>−7</sup> [[Henry (unit)|H]]·[[meter|m]]<sup>−1</sup>}},<ref>[http://physics.nist.gov/cuu/Units/ampere.html]</ref> leading to the consequential defined value for the [[electric constant]] (''ε''<sub>0</sub>) as {{nowrap|''ε''<sub>0</sub> {{=}} 1/(''μ''<sub>0</sub>''c''<sup>2</sup>) ≈ {{val|8.854187817|e=-12|u=[[Farad|F]]·[[meter|m]]<sup>−1</sup>}}}}.<ref>http://physics.nist.gov/cgi-bin/cuu/Value?ep0</ref>
In [[Gaussian units|cgs]] units, the unit charge, ''esu of charge'' or [[statcoulomb]], is defined so that this Coulomb constant is 1 and [[dimensionless]].
 
This formula says that the magnitude of the force is [[directly proportional]]&nbsp;to the magnitude of the charges of each object and [[inverse-square law|inversely proportional to the square]] of the distance between them. The exponent in Coulomb's Law has been found to be equal to &minus;2 with precision of at least {{val|2.7|3.1|e=-16}}.<ref>
{{citation
| last = Williams, Faller, Hill
| title = New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass
| url = http://prola.aps.org/abstract/PRL/v26/i12/p721_1
| journal = [[Physical Review Letters]]
| volume = 26 | pages = 721–724 | year = 1971
| doi = 10.1103/PhysRevLett.26.721
}}</ref>
 
Coulomb's law can also be interpreted in terms of [[atomic units]] with the force expressed in [[Hartree]]s per [[Bohr radius]], the charge in terms of the [[elementary charge]], and the distances in terms of the ''Bohr radius''.
 
===Electric field===
{{main|Electric field}}
It follows from the [[Lorentz Force Law]] that the magnitude of the [[electric field]] (''E'') created by a single point charge (''q'') at a certain distance (''r'') is given by:
 
:<math>E = {1 \over 4\pi\varepsilon_0}\frac{q}{r^2}.</math>
 
For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The [[SI]] units of electric field are [[volt]]s per [[meter]] or [[newtons]] per [[coulomb]].
 
==Vector form==
In order to obtain both the magnitude and direction of the force on a charge, <math>q_1</math> at position <math>\mathbf{r}_1</math>, experiencing a field due to the presence of another charge, ''q''<sub>2</sub> at position <math>\mathbf{r}_2</math>, the full [[Euclidean vector|vector]] form of Coulomb's law is required.
 
:<math>\mathbf{F} = {1 \over 4\pi\varepsilon_0}{q_1q_2(\mathbf{r}_1 - \mathbf{r}_2) \over |\mathbf{r}_1 - \mathbf{r}_2|^3} = {1 \over 4\pi\varepsilon_0}{q_1q_2 \over r^2}\mathbf{\hat{r}}_{21},</math>
 
where <math>r</math> is the separation of the two charges. This is simply the scalar definition of Coulomb's law with the direction given by the [[unit vector]], <math>\mathbf{\hat{r}}_{21}</math>, parallel with the line ''from'' charge <math>q_2</math> ''to'' charge <math>q_1</math>.<ref name="uTexas">[http://farside.ph.utexas.edu/teaching/em/lectures/node28.html Coulomb's law], University of Texas</ref>
 
If both charges have the same [[Plus and minus signs|sign]] (like charges) then the [[Scalar multiplication|product]] <math>q_1q_2</math> is positive and the direction of the force on <math>q_1</math> is given by <math>\mathbf{\hat{r}}_{21}</math>; the charges repel each other. If the charges have opposite signs then the product <math>q_1q_2</math> is negative and the direction of the force on <math>q_1</math> is given by <math>-\mathbf{\hat{r}}_{21}</math>; the charges attract each other.
 
===System of discrete charges===
The principle of [[linear superposition]] may be used to calculate the force on a small test charge, <math>q</math>, due to a system of <math>N</math> discrete charges:
 
:<math>\mathbf{F}(\mathbf{r}) = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i(\mathbf{r} - \mathbf{r}_i) \over |\mathbf{r} - \mathbf{r}_i|^3} = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i \over R_i^2}\mathbf{\hat{R}}_i,</math>
 
where <math>q_i</math> and <math>\mathbf{r}_i</math> are the magnitude and position respectively of the <math>i^{th}</math> charge, <math>\mathbf{\hat{R}}_{i}</math> is a unit vector in the direction of <math>\mathbf{R}_{i} = \mathbf{r} - \mathbf{r}_i</math> (a vector pointing from charge <math>q_i</math> to charge <math>q</math>), and <math>R_{i}</math> is the magnitude of <math>\mathbf{R}_{i}</math> (the separation between charges <math>q_i</math> and <math>q</math>).<ref name="uTexas"/>
 
===Continuous charge distribution===
For a charge distribution an [[integral]] over the region containing the charge is equivalent to an infinite summation, treating each [[infinitesimal]] element of space as a point charge <math>dq</math>.
 
For a linear charge distribution (a good approximation for charge in a wire) where <math>\lambda(\mathbf{r^\prime})</math> gives the charge per unit length at position <math>\mathbf{r^\prime}</math>, and <math>dl^\prime</math> is an infinitesimal element of length,
 
:<math>dq = \lambda(\mathbf{r^\prime})dl^\prime</math>.<ref>[http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml Charged rods], PhysicsLab.org</ref>
 
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate [[capacitor]]) where <math>\sigma(\mathbf{r^\prime})</math> gives the charge per unit area at position <math>\mathbf{r^\prime}</math>, and <math>dA^\prime</math> is an infinitesimal element of area,
 
:<math>dq = \sigma(\mathbf{r^\prime})\,dA^\prime.\,</math>
 
For a volume charge distribution (such as charge within a bulk metal) where <math>\rho(\mathbf{r^\prime})</math> gives the charge per unit volume at position <math>\mathbf{r^\prime}</math>, and <math>dV^\prime</math> is an infinitesimal element of volume,
 
:<math>dq = \rho(\mathbf{r^\prime})\,dV^\prime.</math><ref name="uTexas"/>
 
The force on a small test charge <math>q^\prime</math> at position <math>\mathbf{r}</math> is given by
 
:<math>\mathbf{F} = {q^\prime \over 4\pi\varepsilon_0}\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}.</math>
 
===Graphical representation===
Below is a graphical representation of Coulomb's law, when <math>q_1q_2 > 0</math>. The vector <math>\mathbf{F}_1</math> is the force experienced by <math>q_1</math>. The vector <math>\mathbf{F}_2</math> is the force experienced by <math>q_2</math>. Their magnitudes will always be equal. The vector <math>\mathbf{r}_{21}</math> is the displacement vector between two charges (<math> q_1</math> and <math>q_2</math>).
[[Image:Coulombs.png|center|frame|A graphical representation of Coulomb's law.]]
 
==Electrostatic approximation==
In either formulation, Coulomb&rsquo;s law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the [[electrostatics#The electrostatic approximation|electrostatic approximation]]. When movement takes place, [[magnetic field]]s are produced which alter the force on the two objects. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with [[Albert Einstein|Einstein]]&rsquo;s [[theory of relativity]] taken into consideration.
 
==Table of derived quantities==
{| border="1" style="border-collapse: collapse;" cellpadding="15"
| ||Particle property||Relationship||Field property
|-
|-
|Vector quantity||
{| border="0"
|''Force (on 1 by 2)''
|-
|<math>\mathbf{F}_{21}= {1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r^2}\mathbf{\hat{r}}_{21} \ </math>
|}
|<math>\mathbf{F}_{21}= q_1 \mathbf{E}_{21}</math>||
{| border="0"
|''Electric field (at 1 by 2)''
|-
|<math>\mathbf{E}_{21}= {1 \over 4\pi\varepsilon_0}{q_2 \over r^2}\mathbf{\hat{r}}_{21} \ </math>
|}
|-
|Relationship||<math>\mathbf{F}_{21}=-\mathbf{\nabla}U_{21}</math> || ||<math>\mathbf{E}_{21}=-\mathbf{\nabla}V_{21}</math>
|-
|Scalar quantity||
{| border="0"
|''Potential energy (at 1 by 2)''
|-
|<math>U_{21}={1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r} \ </math>
|}
|<math>U_{21}=q_1 V_{21} \ </math>||
{| border="0"
|''Potential (at 1 by 2)''
|-
|<math>V_{21}={1 \over 4\pi\varepsilon_0}{q_2 \over r} </math>
|}
|}
 
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