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{{For|other examples of "work" in physics|Work (physics)}}
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'''Electrical work''' is the [[Work (physics)|work]] done on a charged particle by an [[electric field]]. The equation for 'electrical' work is, naturally, equivalent to that of 'mechanical' work:
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:<math>W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r}</math>
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:''Q'' is the [[electric charge|charge]] of the particle, q, the unit charge
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:''E'' is the [[electric field]], which at a location is the force at that location divided by a unit ('test') charge
:''F<sub>E</sub>'' is the Coulomb (electric) force
:''r'' is the [[displacement (vector)|displacement]]
:''<math>\cdot</math>'' is the [[dot product]]


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Under normal circumstances, positively charged particles that are free to move will always tend to shift towards the direction of lower voltages (net negative charge). In contrast to this behavior of positive charges, particles that are negatively charged tend to shift towards regions that possess higher voltage (net positive charge).
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However, it requires positive external work to move a positive charge into a region possessing a higher value of [[volt|voltage]]. This is because external work is needed to be done against the [[electric field|field]] of the electric force; The work done here is equal to the work done by the electric field if it were to carry said positive charge the same distance in the opposite direction. Similarly, it requires positive external work to transfer a negatively charged particle from a region of higher voltage to a region of lower voltage.
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The electric force is a [[conservative force]]: work which is done by an electric field is independent of the [[Path_(topology)|path]] taken by the charge. There is no change in the voltage ([[electric potential]]) around '''any''' closed path; when returning to the starting point in a closed path, the net of the external work done is zero. The same phenomenon holds good for electric fields as well.
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The fact explained above forms the basis of one of the most fundamental laws governing electrical and electronic circuits, the [[Kirchhoff's Law|Kirchhoff's voltage law]]. According to this principle, the voltage gains and the drops that occur around any electrical circuit loop is always equal to zero.
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{{Main|Work (physics)}}
 
There is a charged object in empty space, Q+. To move q+ (with the same charge) ''closer'' to Q+ (starting from infinity, where the [[potential energy]]=0, for convenience), positive work would be performed. Mathematically:
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:<math>-\frac{\partial U}{\partial \mathbf{r}} = \mathbf{F}</math>  
 
In this case, U is the [[potential energy]] of q+. So, integrating and using [[Coulomb's Law]] for the force:
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:<math>U = -\int_{r_0}^{r} \mathbf{F} \cdot \, d \mathbf{r}= -\int_{r_0}^{r} \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\mathbf{r^2}} \cdot \, d \mathbf{r}= \frac{q_1q_2}{4\pi\varepsilon_0}(\frac{1}{r_0}- \frac{1}{r})+c</math>
 
c is usually set to 0 and r(0) to infinity (making the 1/r(0) term=0)
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Now, use the relationship
 
:<math> W = -\Delta U \!</math>
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To show that in this case if we start at infinity and move the charge to r,
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:<math>W= \frac{q_1q_2}{4\pi\varepsilon_0}\frac{1}{r}</math>
This could have been obtained equally by using the definition of W and integrating F with respect to r, which will ''prove'' the above relationship.


Note that although in the example both charges are positive, this equation is applicable to any charge configuration (as the product of the charges will be either positive or negative according to their (dis)similarity).
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If one of the charges were to be negative in the earlier example, the work taken to wrench that charge away to infinity would be exactly the same as the work needed in the earlier example to push that charge back to that same position.
 
This is easy to see mathematically, as reversing the boundaries of integration reverses the sign.
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====Uniform electric field====
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Where the  electric field is constant (i.e. ''not'' a function of displacement, r), the work equation simplifies to:


:<math>W =
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(\mathbf{E} \cdot \, \mathbf{r})=\mathbf{F_E} \cdot \, \mathbf{r}</math>
  <li>[http://yirenyiren.com/forum.php?mod=viewthread&tid=743673 http://yirenyiren.com/forum.php?mod=viewthread&tid=743673]</li>
or 'force times distance' (times the cosine of the angle between them).
 
 
  <li>[http://ierode.com/index.php?page=item&id=469839 http://ierode.com/index.php?page=item&id=469839]</li>
==Electric Power==
 
{{Main|Electric power}}
  <li>[http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=959769 http://co.ctbu.edu.cn/forum.php?mod=viewthread&tid=959769]</li>
:<math>P=\frac{\partial W}{\partial t}=\frac{\partial QV}{\partial t}</math>
 
V is the [[voltage]]. Work is defined by:
  <li>[http://web.fytc.bj.cn/news/html/?195717.html http://web.fytc.bj.cn/news/html/?195717.html]</li>
:<math> \delta W = \mathbf{F}\cdot\mathbf{v}\delta t,</math>
 
Therefore
</ul>
:<math>\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v}</math>
 
[[Category:Electromagnetism]]

Revision as of 11:01, 6 February 2014

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