Editing Lorentz force (section)

== Lorentz force and Faraday's law of induction ==
{{main|Faraday's law of induction}}
[[File:Lorentz force - mural Leiden 1, 2016.jpg|upright=1.35|thumb|Lorentz force image on a wall in Leiden]]
Given a loop of wire in a [[magnetic field]], Faraday's law of induction states the induced [[electromotive force]] (EMF) in the wire is:
<math display="block">\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}</math>
where
<math display="block"> \Phi_B = \int_{\Sigma(t)} \mathbf{B}(\mathbf{r}, t)\cdot \mathrm{d}\mathbf{A},</math>
is the [[magnetic flux]] through the loop, {{math|'''B'''}} is the magnetic field, {{math|Σ(''t'')}} is a surface bounded by the closed contour {{math|∂Σ(''t'')}}, at time {{mvar|t}}, {{math|d'''A'''}} is an infinitesimal [[vector area]] element of {{math|Σ(''t'')}} (magnitude is the area of an infinitesimal patch of surface, direction is [[orthogonal]] to that surface patch).

The ''sign'' of the EMF is determined by [[Lenz's law]]. Note that this is valid for not only a ''stationary'' wire{{snd}}but also for a ''moving'' wire.

From [[Faraday's law of induction]] (that is valid for a moving wire, for instance in a motor) and the [[Maxwell Equations]], the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the [[Maxwell Equations]] can be used to derive the [[Faraday's law of induction|Faraday Law]].

Let {{math|∂Σ(''t'')}} be the moving wire, moving together without rotation and with constant velocity {{math|'''v'''}} and {{math|Σ(''t'')}} be the internal surface of the wire. The EMF around the closed path {{math|∂Σ(''t'')}} is given by:<ref name=Landau>{{cite book | last1=Landau | first1= L. D. | last2= Lifshitz | first2 = E. M. | last3 = Pitaevskiĭ | first3 = L. P. | title=Electrodynamics of continuous media |volume=8 |series=Course of Theoretical Physics | year= 1984 | at =§63 (§49 pp. 205–207 in 1960 edition) | edition=2nd | publisher=Butterworth-Heinemann | location=Oxford | isbn=0-7506-2634-8 | url=http://worldcat.org/search?q=0750626348&qt=owc_search}}</ref>
<math display="block">\mathcal{E} = \oint_{\partial \Sigma (t)} \frac{\mathbf{F}}{q}\cdot \mathrm{d} \boldsymbol{\ell} </math>
where <math>\mathbf{E}'(\mathbf{r}, t) =  \mathbf{F}/q(\mathbf{r}, t)</math> is the electric field and {{math|d'''ℓ'''}} is an [[infinitesimal]] vector element of the contour {{math|∂Σ(''t'')}}.{{sfn|Jackson|1998|p=209}}<ref group=nb>Both {{math|d'''''ℓ'''''}} and {{math|d'''A'''}} have a sign ambiguity; to get the correct sign, the [[right-hand rule]] is used, as explained in the article [[Kelvin–Stokes theorem]].</ref> Equating both integrals leads to the field theory form of Faraday's law, given by:{{sfn|Jackson|1998|pp=209-210}}
<math display="block"> \mathcal{E} = \oint_{\partial \Sigma(t)}\mathbf{E}'(\mathbf{r}, t)  \cdot \mathrm{d} \boldsymbol{\ell} = - \frac{\mathrm{d} }{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(\mathbf{r},t) \cdot \mathrm{d} \mathbf{A}. </math>
This result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called the (integral form of) [[Faraday%27s_law_of_induction#Maxwell–Faraday_equation|Maxwell–Faraday equation]]:<ref name=Harrington>{{cite book | first = Roger F. |last=Harrington | author-link = Roger F. Harrington | title = Introduction to electromagnetic engineering | year = 2003 | page = 56 | publisher = Dover Publications | location = Mineola, New York | isbn = 0-486-43241-6 | url = https://books.google.com/books?id=ZlC2EV8zvX8C&q=%22faraday%27s+law+of+induction%22&pg=PA57}}</ref>
<math display="block"> \oint_{\partial \Sigma(t)} \mathbf{E}(\mathbf{r},t) \cdot \mathrm{d} \boldsymbol{\ell}  = - \int_{\Sigma(t)} \frac{\partial \mathbf {B}(\mathbf{r}, t)}{ \partial t } \cdot \mathrm{d} \mathbf{A}.</math>

The two equations are equivalent if the wire is not moving. In case the circuit is moving with a velocity <math>\mathbf{v}</math> in some direction, then, using the [[Leibniz integral rule]] and that {{math|1=div '''B''' = 0}}, gives
<math display="block"> \oint_{\partial \Sigma(t)}\mathbf{E}'(\mathbf{r}, t) \cdot \mathrm{d} \boldsymbol{\ell}= 
- \int_{\Sigma(t)}  \frac{\partial \mathbf{B}(\mathbf{r}, t)}{\partial t} \cdot \mathrm{d}\mathbf{A}  +
\oint_{\partial \Sigma(t)} \left(\mathbf{v} \times \mathbf{B}(\mathbf{r}, t)\right)\cdot \mathrm{d} \boldsymbol{\ell}.
</math>
Substituting the Maxwell-Faraday equation then gives
<math display="block"> \oint_{\partial \Sigma(t)} \mathbf{E}'(\mathbf{r}, t)\cdot \mathrm{d} \boldsymbol{\ell} = 
\oint_{\partial \Sigma(t)} \mathbf{E}(\mathbf{r}, t) \cdot \mathrm{d} \boldsymbol{\ell} +
\oint_{\partial \Sigma(t)} \left(\mathbf{v} \times \mathbf{B}(\mathbf{r}, t)\right) \mathrm{d} \boldsymbol{\ell}
</math>
since this is valid for any wire position it implies that
<math display="block"> \mathbf{F} = q\,\mathbf{E}(\mathbf{r},\, t) + q\,\mathbf{v} \times \mathbf{B}(\mathbf{r},\, t).</math>

Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where Faraday's law is either inadequate or difficult to use, and application of the underlying Lorentz force law is necessary. See [[Faraday paradox#Inapplicability of Faraday's law|inapplicability of Faraday's law]].

If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux {{math|Φ<sub>''B''</sub>}} linking the loop can change in several ways. For example, if the {{math|'''B'''}}-field varies with position, and the loop moves to a location with different B-field, {{math|Φ<sub>''B''</sub>}} will change. Alternatively, if the loop changes orientation with respect to the B-field, the {{math|'''B''' ⋅ d'''A'''}} differential element will change because of the different angle between {{math|'''B'''}} and {{math|d'''A'''}}, also changing {{math|Φ<sub>''B''</sub>}}. As a third example, if a portion of the circuit is swept through a uniform, time-independent {{math|'''B'''}}-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface {{math|∂Σ(''t'')}} time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in {{math|Φ<sub>''B''</sub>}}.

Note that the Maxwell Faraday's equation implies that the Electric Field {{math|'''E'''}} is non conservative when the Magnetic Field {{math|'''B'''}} varies in time, and is not expressible as the gradient of a [[scalar field]], and not subject to the [[gradient theorem]] since its [[Curl (mathematics)|curl]] is not zero.<ref name="Landau"/>{{sfn|Sadiku|2018|pp=424-425}}