Editing Magnetic field (section)

===Faraday's Law===
{{Main|Faraday's law of induction}}

A changing magnetic field, such as a magnet moving through a conducting coil, generates an [[electric field]] (and therefore tends to drive a current in such a coil). This is known as ''Faraday's law'' and forms the basis of many [[electrical generator]]s and [[electric motor]]s. Mathematically, Faraday's law is:
<math display="block">\mathcal{E} = - \frac{\mathrm{d}\Phi}{\mathrm{d}t}</math>

where <math>\mathcal{E}</math> is the [[electromotive force]] (or ''EMF'', the [[voltage]] generated around a closed loop) and {{math|Φ}} is the [[magnetic flux]]—the product of the area times the magnetic field [[Tangential and normal components|normal]] to that area. (This definition of magnetic flux is why {{math|'''B'''}} is often referred to as ''magnetic flux density''.)<ref>{{cite book|last1=Jackson | first1=John David | author-link = John David Jackson (physicist) | title=Classical electrodynamics | date=1975 | publisher=Wiley|location=New York | isbn=9780471431329|edition=2nd|url=https://archive.org/details/classicalelectro00jack_0}}</ref>{{rp|p=210}} The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field that ''opposes'' the ''change'' in the magnetic field that induced it. This phenomenon is known as [[Lenz's law]]. This integral formulation of Faraday's law can be converted<ref group="note" name="ex14">
A complete expression for Faraday's law of induction in terms of the electric {{math|'''E'''}} and magnetic fields can be written as:
<math display="block">\mathcal{E} = - \frac{d\Phi}{dt} = \oint_{\partial \Sigma (t)} \left( \mathbf{E}( \mathbf{r},\ t) +\mathbf{ v} \times \mathbf{B}(\mathbf{r},\ t)\right) \cdot d\boldsymbol{\ell}\ =-\frac {d} {dt} \iint_{\Sigma (t)} d \boldsymbol {A} \cdot \mathbf {B} (\mathbf{r},\ t)</math>
where {{math|'''∂Σ'''(''t'')}} is the moving closed path bounding the moving surface {{math|'''Σ'''(''t'')}}, and {{math|d'''A'''}} is an element of surface area of {{math|'''Σ'''(''t'')}}. The first integral calculates the work done moving a charge a distance {{math|d'''ℓ'''}} based upon the Lorentz force law. In the case where the bounding surface is stationary, the [[Kelvin–Stokes theorem]] can be used to show this equation is equivalent to the Maxwell–Faraday equation.
</ref> into a differential form, which applies under slightly different conditions.

<math display="block">
  \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}
</math>