Editing Magnetic field (section)

==Appearance in Maxwell's equations==
{{Main|Maxwell's equations}}
{{See also|Electromagnetism}}

Like all vector fields, a magnetic field has two important mathematical properties that relates it to its ''sources''. (For {{math|'''B'''}} the ''sources'' are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up ''Maxwell's Equations''. Maxwell's Equations together with the Lorentz force law form a complete description of [[Classical electromagnetism|classical electrodynamics]] including both electricity and magnetism.

The first property is the [[divergence]] of a vector field {{math|'''A'''}}, {{math|'''∇''' · '''A'''}}, which represents how {{math|'''A'''}} "flows" outward from a given point. As discussed above, a {{math|'''B'''}}-field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of {{math|'''B'''}} is zero. (Such vector fields are called [[solenoidal vector field]]s.) This property is called [[Gauss's law for magnetism]] and is equivalent to the statement that there are no isolated magnetic poles or [[magnetic monopole]]s.

The second mathematical property is called the [[curl (mathematics)|curl]], such that {{math|'''∇''' × '''A'''}} represents how {{math|'''A'''}} curls or "circulates" around a given point. The result of the curl is called a "circulation source". The equations for the curl of {{math|'''B'''}} and of {{math|'''E'''}} are called the [[Ampère–Maxwell equation]] and [[Faraday's law of induction|Faraday's law]] respectively.

===Gauss' law for magnetism===
{{main|Gauss's law for magnetism}}

One important property of the {{math|'''B'''}}-field produced this way is that magnetic {{math|'''B'''}}-field lines neither start nor end (mathematically, {{math|'''B'''}} is a [[solenoidal vector field]]); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path.<ref name="lieb">{{cite journal | last1=Lieberherr | first1=Martin | date=6 July 2010|title=The magnetic field lines of a helical coil are not simple loops | journal=American Journal of Physics|volume=78|issue=11|pages=1117–1119|bibcode=2010AmJPh..78.1117L|doi=10.1119/1.3471233|doi-access=free}}</ref> Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet {{math|'''B'''}}-field lines continue through the magnet from the south pole back to the north.<ref group="note" name="ex08">To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction.</ref> If a {{math|'''B'''}}-field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point.

More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number"<ref group="note" name="ex09">As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total "number" of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead.</ref> of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to [[Gauss's law for magnetism]]:
<math display="block">\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
where the integral is a [[surface integral]] over the [[closed surface]] {{math|''S''}} (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since {{math|d'''A'''}} points outward, the dot product in the integral is positive for {{math|'''B'''}}-field pointing out and negative for {{math|'''B'''}}-field pointing in.

===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>

===Ampère's Law and Maxwell's correction===
{{Main|Ampère's circuital law}}

Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as ''Maxwell's correction to Ampère's law'' and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.)

The full law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used.

The Maxwell term ''is'' critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to form [[electromagnetic waves]], such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again. These, though, are usually described using the differential form of this equation given below.

<math display="block">
  \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}
</math>

where {{math|'''J'''}} is the complete microscopic [[current density]], and {{mvar|ε{{sub|0}}}} is the [[vacuum permittivity]].

As discussed above, materials respond to an applied electric {{math|'''E'''}} field and an applied magnetic {{math|'''B'''}} field by producing their own internal "bound" charge and current distributions that contribute to {{math|'''E'''}} and {{math|'''B'''}} but are difficult to calculate. To circumvent this problem, {{math|'''H'''}} and {{math|'''D'''}} fields are used to re-factor Maxwell's equations in terms of the ''free current density'' {{math|'''J'''<sub>f</sub>}}:

<math display="block">\nabla \times \mathbf{H} = \mathbf{J}_\mathrm{f} + \frac{\partial \mathbf{D}} {\partial t}</math>

These equations are not any more general than the original equations (if the "bound" charges and currents in the material are known). They also must be supplemented by the relationship between {{math|'''B'''}} and {{math|'''H'''}} as well as that between {{math|'''E'''}} and {{math|'''D'''}}. On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents.