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In [[physics]], more particularly in [[electrodynamics]], the law first formulated by [[Jean-Baptiste Biot]] and [[Felix Savart|Félix Savart]] <ref>J.-B. Biot and F. Savart, ''Note sur le Magnétisme de la pile de Volta,'' Annales Chim. Phys. vol. '''15''', pp. 222-223 (1820)</ref> describes the [[magnetic induction]] '''B''' (proportional to the magnetic field '''H''') caused by a direct electric current in a wire. Biot and Savart interpreted their measurements by an integral relation. [[Laplace]] gave a differential form of their result, which now often is also referred to  as the Biot-Savart law, or sometimes as the Biot-Savart-Laplace law. By integrating Laplace's equation over an infinitely long wire, the original integral form of Biot and Savart is obtained.
#REDIRECT [[Biot–Savart law]]
[[Image:Laplace magnetic.png|right|thumb|250px|Magnetic induction d'''B''' at point '''r''' due to  infinitesimal piece d'''s''' (red) of wire (blue) transporting electric current ''i''. ]]
==Laplace's formula==
The infinitesimal magnetic induction <math>\scriptstyle d\vec{\mathbf{B}} </math> at point <math>\scriptstyle \vec{\mathbf{r}} </math> (see figure on the right) is given by the following formula due to Laplace,
:<math>
d\vec{\mathbf{B}} = k \frac{i d\vec{\mathbf{s}} \times \vec{\mathbf{r}}} {|\vec{\mathbf{r}}|^3},
</math>
where the magnetic induction is given as a [[vector product]], i.e., is perpendicular to the plane spanned by <math>\scriptstyle d\vec{\mathbf{s}} </math> and <math>\scriptstyle \vec{\mathbf{r}} </math>.  The electric current ''i'' is constant in time. The  piece of the wire contributing to the magnetic induction can be seen as a vector of infinitesimal length d''s'' and with direction tangent to the  wire. The constant ''k'' depends on the units chosen. In rationalized SI units ''k'' is  the [[magnetic constant]] (vacuum permeability) divided by 4&pi;. The magnetic constant &mu;<sub>0</sub> = 4&pi; &times;10<sup>&minus;7</sup> N/A<sup>2</sup> (newton divided by ampere squared). In Gaussian units ''k'' = 1 / ''c'' (one over the velocity of light). 
 
If we remember the fact that the vector '''r''' has dimension length, we see that this equation is an [[Inverse-square_law|inverse distance squared law]].
 
==Formula of Biot and Savart==
[[Image:Biot Savart.png|left|thumb|250px|Field '''B''' due to current ''i'' in infinitely long straight wire.]]
Take a straight infinitely long wire transporting the current ''i''. Write, using  ''R'' = ''r''sin&alpha; (see the figure),
:<math>
d\vec{\mathbf{s}} \times \vec{\mathbf{r}} = \hat{\mathbf{e}} \,r\sin\alpha\, ds =  \hat{\mathbf{e}}\, R\,ds,
</math>
where <math>\scriptstyle \hat{\mathbf{e}} </math> is a unit vector perpendicular to the plane spanned by the wire and the vector <math>\scriptstyle \vec{\mathbf{R}}</math> perpendicular to the wire. Note that if <math>\scriptstyle d\vec{\mathbf{s}} </math> moves along the wire all contributions from the segments to the magnetic induction are along this unit vector.  Hence, if we integrate over the wire we add up all these contributions, so that
:<math>
|\vec{\mathbf{B}}| = i R k \int_{-\infty}^{\infty} \frac{ds}{(s^2+R^2)^{3/2}}
</math>
where, by the [[Pythagorean theorem]],
:<math>
|\vec{\mathbf{r}}|^2 = s^2 + R^2.
</math>
Substition of ''y'' = ''s'' / R and ''y'' = cot&phi; = cos&phi; / sin&phi;, successively, gives
:<math>
|\vec{\mathbf{B}}| = \frac{ik}{R} \int_{-\infty}^{\infty} \frac{dy}{(y^2+1)^{3/2}} =
\frac{ik}{R} \int_{0}^{\pi} \sin\phi \, d\phi = \frac{2 ik}{R},
</math>
where ''i'' is the current and ''R'' the distance of the point of observation of the magnetic induction to the wire. The constant ''k'' depends on the choice of electromagnetic units and is 10<sup>&minus;7</sup> henry/m [= Vs/A = N/A<sup>2</sup>] in rationalized [[SI]] units. This equation gives the original formulation of Biot and Savart. The SI dimension of ''B'' is T [tesla: 1 T =  1 N/(Am), newton divided by ampere meter]. A field of 1 T (SI) corresponds to 10000 gauss (cgs units).
==Generalization to current density==
:''From hereon vectors are indicated by bold letters, arrows on top are omitted''.
 
In the above we wrote ''i'' for the current, which is equal to the current density ''J'' times the cross section ''A'' of the wire. If the current density ''J'' is not constant over the cross section, i.e., ''J'' = ''J''(<b>r</b>' ), we must use an integral over the cross section. Rather than introducing a surface element we multiply immediately by d''s'' and obtain an infinitesimal volume element <math>\scriptstyle d\mathbf{r}'</math>,
:<math>
i d\mathbf{s} = \iint_{A} \mathbf{J}(\mathbf{r}') d\mathbf{r}',
</math> 
where we assumed that the current density is a vector parallel to the segment d'''s''' and where
the volume element has height d''s'' and  base an infinitesimal element of ''A''. The '''B'''-field at point '''r''', due to a volume ''V'' = ''As'' of the wire becomes,
:<math>
\mathbf{B}(\mathbf{r}) = k \iiint_{V} \frac{\mathbf{J}(\mathbf{r}') \times (\mathbf{r}-\mathbf{r'})}
{|\mathbf{r}-\mathbf{r'}|^3}d\mathbf{r}' .
</math>
Note that:
:<math>
\frac{\mathbf{r}-\mathbf{r'}}{|\mathbf{r}-\mathbf{r'}|^3} = \boldsymbol{\nabla} \left( \frac{1}{|\mathbf{r}-\mathbf{r'}|} \right),
</math>
where we choose the nabla operator to act on the ''unprimed'' coordinates and hence it may be moved outside the integral, giving the following ''generalized form of the Biot-Savart law'' for the magnetic induction at point '''r''':
:<math>
\mathbf{B}(\mathbf{r}) = k \boldsymbol{\nabla} \times \iiint_{V} \frac{\mathbf{J}(\mathbf{r}')}  {|\mathbf{r}-\mathbf{r'}|}d\mathbf{r}',
</math>
where ''V'' is a finite volume segment of the current generating the '''B'''-field. The total '''B'''-field is obtained by having ''V'' encompass all current.
 
 
==References==
<references />
Further reading:
* J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, John Wiley, New York (1975).
 
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