Editing Electromotive force (section)

==Formal definitions==
''Inside'' a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal ''N'' and the positive terminal ''P''.
This leads to an [[electrostatic field]] <math>\boldsymbol{E}_\mathrm{open\ circuit}</math> that points from ''P'' to ''N'', whereas the emf of the source must be able to drive current from ''N'' to ''P'' when connected to a circuit.
This led [[Max Abraham]]<ref name=Abraham>
{{cite book
 | title = The Classical Theory of Electricity and Magnetism
 | first1 = M.|last1=Abraham|first2=R.|last2=Becker
 | publisher = Blackie & Son
 | year = 1932
 | pages = 116–122
 | url = https://pdfcoffee.com/abraham-max-amp-becker-richard-classical-theory-of-electricity-and-magnetism-1932-pdf-pdf-free.html
 }}</ref> to introduce the concept of a [[Nonelectrostatic electric fields|nonelectrostatic field]] <math>\boldsymbol{E}'</math> that exists only inside the source of emf.
In the open-circuit case, <math>\boldsymbol{E}' = - \boldsymbol{E}_\mathrm{open\ circuit}</math>, while when the source is connected to a circuit the electric field <math>\boldsymbol{E}</math> inside the source changes but <math>\boldsymbol{E}'</math> remains essentially the same.
In the open-circuit case, the [[Conservative field|conservative]] electrostatic field created by separation of charge exactly cancels the forces producing the emf.<ref name=Griffiths>
{{cite book
 | title = Introduction to Electrodynamics
 | first = David J|last=Griffiths
 | publisher = Pearson/Addison-Wesley
 | year = 1999
 | isbn = 978-0-13-805326-0
 | page = [https://archive.org/details/introductiontoel00grif_0/page/293 293]
 | edition = 3rd
 | url = https://archive.org/details/introductiontoel00grif_0/page/293
 }}</ref> 
Mathematically:

<math display="block">\mathcal{E}_\mathrm{source}  = \int_{N}^{P} \boldsymbol{E}' \cdot \mathrm{d} \boldsymbol{ \ell }
= - \int_{N}^{P} \boldsymbol{E}_\mathrm{open\ circuit} \cdot \mathrm{d} \boldsymbol{ \ell }
=V_P - V_N \ ,</math>

where <math>\boldsymbol{E}_\mathrm{open\ circuit}</math> is the conservative electrostatic field created by the charge separation associated with the emf, <math>\mathrm{d}\boldsymbol{\ell}</math> is an element of the path from terminal ''N'' to terminal ''P'', '<math>\cdot</math>' denotes the vector [[dot product]], and <math>V</math> is the electric scalar potential.<ref name="diode">
Only the electric field that results from charge separation caused by the emf is counted. While a solar cell has an electric field that results from a contact potential (see [[Electromotive force#Contact potentials|contact potentials]] and [[Electromotive force#Solar cell|solar cells]]), this electric field component is not included in the integral. Only the electric field that results from charge separation caused by photon energy is included.
</ref> 
This emf is the work done on a unit charge by the source's nonelectrostatic field <math>\boldsymbol{E}'</math> when the charge moves from ''N'' to ''P''.

When the source is connected to a load, its emf is just
<math>\mathcal{E}_\mathrm{source} = \int_{N}^{P} \boldsymbol{E}' \cdot \mathrm{d} \boldsymbol{ \ell}\ ,</math>
and no longer has a simple relation to the electric field <math>\boldsymbol{E}</math> inside it.

In the case of a closed path in the presence of a varying [[magnetic field]], the integral of the [[electric field]] around the (stationary) closed loop <math>C</math> may be nonzero.
Then, the "''induced emf''" (often called the "induced voltage") in the loop is:<ref>{{cite book
 | title = Beyond the mechanical universe: from electricity to modern physics
 |first1=Richard P.|last1=Olenick |first2=Tom M.|last2=Apostol |first3=David L.|last3=Goodstein 
 | publisher = Cambridge University Press
 | year = 1986
 | isbn = 978-0-521-30430-6
 | page = 245
 | url = https://books.google.com/books?id=Ht4T7C7AXZIC&q=define+electromotive-force+around-a-closed-path&pg=RA1-PA245
 }}</ref>

<math display="block">\mathcal{E}_C = \oint_{C} \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{ \ell }
= - \frac{d\Phi_C}{dt}
= - \frac{d}{dt} \oint_{C} \boldsymbol{A} \cdot \mathrm{d} \boldsymbol{ \ell }\ ,
</math>

where <math>\boldsymbol{E}</math> is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary, but stationary, closed curve <math>C</math> through which there is a time-varying [[magnetic flux]] <math>\Phi_C</math>, and <math>\boldsymbol{A}</math> is the [[vector potential]].
The electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is [[Conservative force|conservative]] (i.e., the work done against the field around a closed path is zero, see [[Kirchhoff's circuit laws#Kirchhoff's voltage law|Kirchhoff's voltage law]], which is valid, as long as the circuit elements remain at rest and radiation is ignored<ref name="McDonald">
{{cite web 
|last1=McDonald |first1=Kirk T. 
|title=Voltage Drop, Potential Difference and EMF 
|url=http://kirkmcd.princeton.edu/examples/volt.pdf 
|website=Physics Examples 
|publisher=Princeton University
|date=2012
}}</ref>).
That is, the "induced emf" (like the emf of a battery connected to a load) is not a "voltage" in the sense of a difference in the electric scalar potential.

If the loop <math>C</math> is a conductor that carries current <math>I</math> in the direction of integration around the loop, and the magnetic flux is due to that current, we have that <math>\Phi_B = L I</math>, where <math>L</math> is the self inductance of the loop.
If in addition, the loop includes a coil that extends from point 1 to 2, such that the magnetic flux is largely localized to that region, it is customary to speak of that region as an [[inductor]], and to consider that its emf is localized to that region.
Then, we can consider a different loop <math>C'</math> that consists of the coiled conductor from 1 to 2, and an imaginary line down the center of the coil from 2 back to 1.  
The magnetic flux, and emf, in loop <math>C'</math> is essentially the same as that in loop <math>C</math>:<math display="block">\mathcal{E}_C = \mathcal{E}_{C'}
= - \frac{d\Phi_{C'}}{dt}
= - L \frac{d I}{dt}
= \oint_C \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{ \ell }
= \int_1^2 \boldsymbol{E}_\mathrm{conductor} \cdot \mathrm{d} \boldsymbol{ \ell }
- \int_1^2 \boldsymbol{E}_\mathrm{center\ line} \cdot \mathrm{d} \boldsymbol{ \ell }\ .
</math>

For a good conductor, <math>\boldsymbol{E}_\mathrm{conductor}</math> is negligible, so we have, to a good approximation,
<math display="block">L \frac{d I}{dt}
= \int_1^2 \boldsymbol{E}_\mathrm{center\ line} \cdot \mathrm{d} \boldsymbol{ \ell } 
= V_1 - V_2\ ,
</math>
where <math>V</math> is the electric scalar potential along the centerline between points 1 and 2.

Thus, we can associate an effective "voltage drop" <math>L\ d I / d t</math> with an inductor (even though our basic understanding of induced emf is based on the vector potential rather than the scalar potential), and consider it as a load element in Kirchhoff's voltage law,

<math display="block"> \sum \mathcal{E}_\mathrm{source} = \sum_\mathrm{load\ elements} \mathrm{voltage\ drops},
</math>

where now the induced emf is not considered to be a source emf.<ref name="FeynmanCook2">{{cite book
 | title = The Feynman Lectures on Physics, Vol. II, chap. 22
 | first1 = R.P.|last1=Feynman|first2=R.B.|last2=Leighton|first3=M.|last3=Sands
 | publisher = Addison Wesley
 | year = 1964
 | url = https://www.feynmanlectures.caltech.edu/II_22.html }}</ref>

This definition can be extended to arbitrary sources of emf and paths ''<math>C</math>'' moving with velocity <math>\boldsymbol{v}</math> through the electric field <math>\boldsymbol{E}</math> and magnetic field <math>\boldsymbol{B}</math>:<ref name=Cook2>{{cite book
 | title = The Theory of the Electromagnetic Field
 | first = David M.|last=Cook
 | publisher = Courier Dover
 | year = 2003
 | isbn = 978-0-486-42567-2
 | page = 158
 | url = https://books.google.com/books?id=bI-ZmZWeyhkC&pg=PA158
 }}</ref>

<math display="block">\begin{align}
\mathcal{E} &= \oint_{C} \left[\boldsymbol{E}  + \boldsymbol{v} \times \boldsymbol{B} \right] \cdot \mathrm{d} \boldsymbol{ \ell } \\
&\qquad+\frac{1}{q}\oint_{C}\mathrm {Effective \ chemical \ forces \ \cdot} \ \mathrm{d} \boldsymbol{ \ell } \\
&\qquad\qquad+\frac{1}{q}\oint_{C}\mathrm { Effective \ thermal \ forces\ \cdot}\  \mathrm{d} \boldsymbol{ \ell } \ ,
\end{align} </math>

which is a conceptual equation mainly, because the determination of the "effective forces" is difficult.
The term <math> \oint_{C} \left[\boldsymbol{E}  + \boldsymbol{v} \times \boldsymbol{B} \right] \cdot \mathrm{d} \boldsymbol{ \ell } </math>
is often called a "motional emf".