101.229.56.50: /* Dispersion relation */

2012-06-10T12:52:24Z

Dispersion relation

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'''Coulomb's constant''', the '''electric force constant''', or the '''electrostatic constant''' (denoted {{math|{{SubSup|k|e|s=0}}}}) is a [[proportionality constant]] in equations relating electric variables and is exactly equal to {{math|{{SubSup|k|e|s=0}}}} = {{gaps|8.987|551|787|368|1764}}{{e|9}} [[newton (unit)|N]]·[[metre|m]]2/[[coulomb|C]]2 (m/[[farad|F]]). It was named after the [[French people|French]] [[physicist]] [[Charles-Augustin de Coulomb]] (1736–1806).

== Value of the constant ==

The exact value of Coulomb's constant {{math|{{SubSup|k|e|s=0}}}} comes from three of the fundamental, invariant quantities that define [[Vacuum#Electromagnetism|free space]] in the SI system: the speed of light {{SubSup|c|0|s=0}}, magnetic permeability {{SubSup|μ|0|s=0}}, and electric permittivity {{SubSup|ε|0|s=0}}, related by [[Maxwell's equations#Relation between electricity.2C magnetism.2C and the speed of light|Maxwell]] as:

:<math> \frac{1}{\mu_0\varepsilon_0}=c_0^2. </math>

Because of the way the [[SI base unit]] system made the [[Natural units#Electromagnetism units|natural units for electromagnetism]], the [[speed of light in vacuum]] {{SubSup|c|0|s=0}} is {{val|fmt=commas|299792458|u=m [[second|s]]−1}}, the [[magnetic permeability]] {{SubSup|μ|0|s=0}} of free space is {{nowrap|4''π''·10−7 [[Henry (unit)|H]] m−1}}, and the [[electric permittivity]] {{SubSup|ε|0|s=0}} of free space is {{nowrap|1 {{frac}} ({{SubSup|μ|0|s=0}}{{SubSup|c|0|2|s=0}}) ≈ {{val|fmt=commas|8.85418782|e=-12|u=[[Farad|F]] m−1}}}},<ref>[http://physics.nist.gov/cgi-bin/cuu/Value?ep0 CODATA Value: electric constant]. Physics.nist.gov. Retrieved on 2010-09-28.</ref>
so that<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c3 Coulomb's constant], Hyperphysics</ref>
:<math>\begin{align}
k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\cdot10^{-7}\mathrm{H\ m}^{-1}\\
&= 8.987\ 551\ 787\ 368\ 176\ 4\cdot10^9\mathrm{N\ m^2\ C}^{-2}.
\end{align}</math>

== Use of Coulomb's constant ==

Coulomb's constant is used in many electric equations, although it is sometimes expressed as the following product of the [[Vacuum permittivity]] constant:
:<math>\begin{align}
k_\text{e} &= \frac{1}{4\pi\varepsilon_0}
\end{align}</math>.

Some examples of use of Coulomb's constant are the following:

[[Coulomb's law]]:
:<math>|\boldsymbol{F}|=k_\text{e}{|q_1q_2|\over r^2}</math>.

[[Electric potential energy]]:
:<math> U_\text{E}(r) = k_\text{e}\frac{qQ}{r}</math>.

[[Electric field]]:
:<math>\mathbf{E} = k_\text{e} \sum_{i=1}^N \frac{Q_i}{r_i^2} \mathbf{\hat{r}}_i</math>.

== See also ==
* [[Vacuum permittivity]]
* [[Vacuum permeability]]

== References ==

{{Reflist}}

[[Category:Electricity]]
[[Category:Electromagnetism]]
[[Category:Physical constants]]

Two-stream instability - Revision history

101.229.56.50: /* Dispersion relation */