Editing Magnetic field (section)

===Magnetic field due to moving charges and electric currents===
{{Main|Electromagnet|Biot–Savart law|Ampère's circuital law|l3=Ampère's law}}
[[Image:Manoderecha.svg|thumb|right|[[Right hand grip rule]]: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.]]

All moving charged particles produce magnetic fields. Moving [[point particle|point]] charges, such as [[electron]]s, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.<ref>{{harvnb|Griffiths|1999|p=438}}</ref>

Magnetic field lines form in [[concentric]] circles around a [[Cylinder (geometry)|cylindrical]] current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "[[right-hand grip rule]]" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.)
[[File:VFPt cylindrical tightly-wound coil-and-bar-magnet-comparison stacked.svg|150 px|thumb|left|A [[Solenoid]] with electric current running through it behaves like a magnet.]]

Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "[[solenoid]]" enhances this effect. A device so formed around an iron [[Magnetic core|core]] may act as an ''electromagnet'', generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil.

The magnetic field generated by a steady current {{math|I}} (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point){{refn|group="note" |name="ex12"|In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.<ref name="Griffiths4ed"/>{{rp|p=223}}}} is described by the ''[[Biot–Savart law]]'':<ref name="Griffiths4ed">{{cite book |last1=Griffiths |first1=David J. | author-link = David J. Griffiths |title=[[Introduction to Electrodynamics]] |date=2017 |publisher=Cambridge University Press | isbn=9781108357142 |edition=4th}}</ref>{{rp|p=224}}
<math display="block"> \mathbf{B} = \frac{\mu_0I}{4\pi}\int_{\mathrm{wire}}\frac{\mathrm{d}\boldsymbol{\ell} \times \mathbf{\hat r}}{r^2},</math>
where the integral sums over the wire length where vector {{math|d'''ℓ'''}} is the vector [[line element]] with direction in the same sense as the current {{math|''I''}}, {{math|''μ''<sub>0</sub>}} is the [[magnetic constant]], {{math|''r''}} is the distance between the location of {{math|d'''ℓ'''}} and the location where the magnetic field is calculated, and {{math|'''r̂'''}} is a unit vector in the direction of {{math|'''r'''}}. For example, in the case of a sufficiently long, straight wire, this becomes:
<math display="block"> |\mathbf{B}| = \frac{\mu_0}{2\pi r}I</math>
where {{math|1=''r'' = {{abs|'''r'''}}}}. The direction is tangent to a circle perpendicular to the wire according to the right hand rule.<ref name="Griffiths4ed"/>{{rp|p=225}}

A slightly more general<ref>{{harvnb|Griffiths|1999|pp=222–225}}</ref><ref group="note" name="ex13">
The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of {{math|'''B'''}} being zero, which is always valid.  (There are no magnetic charges.)</ref> way of relating the current <math>I</math> to the {{math|'''B'''}}-field is through [[Ampère's circuital law|Ampère's law]]:
<math display="block">\oint \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}},</math>
where the [[line integral]] is over any arbitrary loop and <math>I_\text{enc}</math> is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the {{math|'''B'''}}-field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

In a modified form that accounts for time varying electric fields, Ampère's law is one of four [[Maxwell's equations]] that describe electricity and magnetism.