Heine–Borel theorem: Difference between revisions

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At the end of the article was the statement that one direction of the proof of the Heine-Borel theorem is equivalent to the Ultrafilter Principle. The citation given (Schechter) states that this is true for the filter version. I corrected this.
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→‎Proof: Remove circular appeal to Cantor's Intersection Theorem, just need a Cauchy sequence.
 
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{{Distinguish|Electric potential energy}}
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{{Electromagnetism|cTopic=[[Electrical network|Electrical Network]]}}
{{More footnotes|date=April 2008}}
In [[classical electromagnetism]], the '''electric  potential''' (a [[scalar (physics)|scalar]] quantity denoted by {{math|Φ}}, {{math|Φ<sub>'''''E'''''</sub>}} or {{math|''V''}} and also called the ''electric field potential'' or the ''electrostatic potential'') at a point of space is the amount of [[electric potential energy]] that a unitary point charge would have when located at that point.
 
The electric potential at a point is equal to the [[electric potential energy]] (measured in [[joule]]s) of any charged particle at that location divided by the [[electric charge|charge]] (measured in [[coulomb]]s) of the particle. Since the charge of the test particle has been divided out, the electric potential is a "property" related only to the electric field itself and not the test particle. The electric potential can be calculated at a point in either a static (time-invariant) [[electric field]] or in a dynamic (varying with time) [[electric field]] at a specific time, and has the units of [[joules]] per [[coulomb]] ({{math|''J C''<sup>–1</sup>}}), or [[volt]]s ({{math|''V''}}).
 
There is also a generalized electric [[scalar potential]] that is used in [[electrodynamics]] when time-varying electromagnetic fields are present. This generalized electric potential cannot be simply interpreted as the ratio of potential energy to charge, however.
 
== Introduction ==
Objects may possess a property known as an [[electric charge]].  An [[electric field]] exerts a force on charged objects. If the charged object has a positive charge, the force will be in the direction of the [[electric field vector]] at that point. The force will be in the opposite direction if the charge is negative.  The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector.  A net force acting on an object will cause it to [[acceleration|accelerate]], as explained by [[Classical mechanics]] which explores concepts such as [[Force (physics)|force]], [[energy]], [[potential]] etc. The electric potential (or simply potential) at a point in an electric field is defined as the work done in moving a unit positive charge from infinity to that point. The electric potential at infinity is assumed to be zero.
 
Force and potential energy are directly related. As an object moves in the direction that the force accelerates it, its potential energy decreases. For example, the gravitational potential energy of a cannonball at the top of a hill is greater than at the base of the hill. As the object falls, that potential energy decreases and is translated to motion, or inertial (kinetic) energy.
 
For certain force fields, it is possible to define the "potential" of a field such that the potential energy of an object due to a field depends only on the position of the object with respect to the field. Those forces must affect objects depending only on the intrinsic properties of the object (e.g., [[mass]] or charge) and the position of the object, and obey certain other mathematical rules.
 
Two such force fields are the gravitational force field ([[gravity]]) and the electric field in the absence of time-varying magnetic fields. The potential of an electric field at a point is called the electric potential.  The synonymous term "electrostatic potential" is also in common use.
 
The electric potential and the [[magnetic vector potential]] together form a [[four vector]], so that the two kinds of potential are mixed under [[Lorentz transformation]]s.
 
== In electrostatics ==
The electric potential at a point '''r''' in a static electric field '''E''' is given by the [[line integral]]
 
{{Equation box 1
|indent=:
|equation=<math>V_\mathbf{E} = - \int_C \mathbf{E} \cdot \mathrm{d} \boldsymbol{\ell} \,</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
 
where ''C'' is an arbitrary path connecting the point with zero potential to '''r'''. When the [[Curl (mathematics)|curl]] {{nowrap|'''&nabla;''' &times; '''E'''}} is zero, the line integral above does not depend on the specific path ''C'' chosen but only on its endpoints. In this case, the electric field is [[conservative vector field|conservative]] and determined by the [[gradient]] of the potential:
 
{{Equation box 1
|indent=:
|equation=<math>\mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}. \, </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
 
Then, by [[Gauss's law]], the potential satisfies [[Poisson's equation]]:
:<math>\mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} V_\mathbf{E} \right ) = -\nabla^2 V_\mathbf{E} = \rho / \varepsilon_0, \, </math>
 
where ''ρ'' is the total [[charge density]] (including [[bound charge]]) and '''∇'''· denotes the [[divergence]].
 
The concept of electric potential  is closely linked with [[potential energy]]. A [[test charge]] ''q'' has an [[electric potential energy]] ''U''<sub>'''E'''</sub> given by
 
:<math>U_ \mathbf{E} = q\,V. \, </math>
 
The potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero.
 
These equations cannot be used if the curl {{nowrap|'''&nabla;''' &times; '''E''' ≠ 0}}, i.e., in the case of a ''nonconservative electric field'' (caused by a changing [[magnetic field]]; see [[Maxwell's equations]]).  The generalization of electric potential to this case is described below.
 
=== Electric potential due to a point charge ===
[[File:Electric potential varying charge.gif|right|thumb|150x150px|The electric potential created by a charge ''Q'' is ''V''=''Q''/(4πε<sub>o</sub>''r''). Different values of ''Q'' will make different values of electric potential ''V'' (shown in the image).]]
 
The electric potential created by a point charge ''Q'', at a distance ''r'' from the charge (relative to the potential at infinity), can be shown to be
 
:<math> V_\mathbf{E} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}, \, </math>
 
where ''ε''<sub>0</sub> is the [[electric constant]] (permitivity of free space). This is known as the Coulomb potential.
 
The electric potential due to a system of point charges is equal to the sum of the point charges' individual potentials.  This fact simplifies calculations significantly, since addition of potential (scalar) fields is much easier than addition of the electric (vector) fields.
 
The equation given above for the electric potential (and all the equations used here) are in the forms required by [[SI units]]. In some other (less common) systems of units, such as [[Gaussian units|CGS-Gaussian]], many of these equations would be altered.
 
== Generalization to electrodynamics ==
 
When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential ''V'' because the electric field is no longer [[conservative force|conservative]]: <math>\textstyle\int_C \mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}</math> is path-dependent because <math>\mathbf{\nabla} \times \mathbf{E} \neq \mathbf{0} </math> ([[Faraday's law of induction]]).
 
Instead, one can still define a scalar potential by also including the [[magnetic vector potential]] '''A'''.  In particular, '''A''' is defined to satisfy:
 
:<math>\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}, \, </math>
 
where '''B''' is the [[magnetic field]]. Because [[Gauss's law for magnetism|the divergence of the magnetic field is always zero]] due to the absence of [[magnetic monopole]]s, such an '''A''' can always be found. Given this, the quantity
:<math>\mathbf{F} = \mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}</math>
''is'' a conservative field by [[Faraday's law of induction|Faraday's law]] and one can therefore write
:<math>\mathbf{E} = -\mathbf{\nabla}V - \frac{\partial\mathbf{A}}{\partial t}, \, </math>
 
where ''V'' is the scalar potential defined by the conservative field '''F'''.
 
The electrostatic potential is simply the special case of this definition where '''A''' is time-invariant.  On the other hand, for time-varying fields, note that
:<math>-\int_a^b \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} \neq V_{(b)} - V_{(a)}, \, </math>
unlike electrostatics.
 
Note that this definition of ''V'' depends on the [[gauge fixing|gauge choice]] for the vector potential '''A''' (the [[gradient]] of any scalar field can be added to '''A''' without changing '''B'''). One choice is the [[Coulomb gauge]], in which we choose {{nowrap|'''&nabla;''' · '''A''' {{=}} 0}}.  In this case, we obtain
:<math>-\nabla^2 V = \rho/\varepsilon_0, \, </math>
where ''ρ'' is the [[charge density]], just as for electrostatics. Another common choice is the [[Lorenz gauge]], in which we choose '''A''' to satisfy
:<math>\mathbf{\nabla} \cdot \mathbf{A} = - \frac{1}{c^2} \frac{\partial V}{\partial t}. \, </math>
 
==Units==
The [[SI]] unit of electric potential is the [[volt]] (in honor of [[Alessandro Volta]]), which is why a difference in electric potential between two points is known as [[voltage]]. Older units are rarely used today.  Variants of the [[centimeter gram second system of units]] included a number of different units for electric potential, including the [[Conversion of units#Electromotive_force.2C_electric_potential_difference|abvolt]] and the [[statvolt]].
 
==Galvani potential versus electrochemical potential==
{{main|Galvani potential|Electrochemical potential|Fermi level}}
Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a [[voltmeter]] is connected between two different types of metal, it measures not the electric potential difference, but instead the potential difference corrected for the different atomic environments.<ref>{{cite book|url=http://books.google.com/books?id=09QI-assq1cC&pg=PA22 |title=Fundamentals of electrochemistry|first= Vladimir Sergeevich|last= Bagotskii|page=22|isbn=978-0-471-70058-6|year=2006}}</ref> The quantity measured by a voltmeter is called [[electrochemical potential]] or [[fermi level]], while the pure unadjusted electric potential is sometimes called [[Galvani potential]]. The terms "voltage" and "electric potential" are a bit ambiguous in that, in practice, they can refer to ''either'' of these in different contexts.
 
==See also==
*[[Absolute electrode potential]]
*[[Electrochemical potential]]
*[[Electrode potential]]
*[[Gluon field]]
*[[Liénard–Wiechert potential]]
*[[Mathematical descriptions of the electromagnetic field]]
*[[Voltage]], or (electric) potential difference
 
==References==
{{reflist}}
*{{cite book |author=Griffiths, David J.|title=Introduction to Electrodynamics| edition=3rd.| publisher=Prentice Hall |year=1998 |isbn=0-13-805326-X}}
*{{cite book|last=Jackson|first=John David|title=Classical Electrodynamics|publisher = John Wiley & Sons, Inc. |year=1999|location=USA|edition=3rd.|isbn=978-0-471-30932-1}}
*{{cite book | author=Wangsness, Roald K.|title=Electromagnetic Fields| publisher=Wiley |year=1986|edition=2nd., Revised, illustrated|isbn=978-0-471-81186-2}}
 
[[Category:Potentials]]
[[Category:Electrostatics]]
[[Category:Physical quantities]]
 
[[de:Elektrostatik#Potential und Spannung]]

Latest revision as of 09:59, 25 October 2014

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