Biot-Savart law: Difference between revisions

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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 field 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 field 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> 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 constant ''k'' depends on the units chosen. In rationalized SI units ''k'' is  the [[magnetic constant]] (vacuum permeability) &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'' = cos&phi;/sin&phi; gives
:<math>
|\vec{\mathbf{B}}| = \frac{ik}{R} \int_{-\infty}^{\infty} \frac{dy}{(y^2+1)^{3/2}} =
\frac{ik}{R} \int_{-\pi}^{\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. This equation gives the original formulation of Biot and Savart.
 
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==References==
<references />

Latest revision as of 12:18, 22 April 2011

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