Biot-Savart law: Difference between revisions

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#REDIRECT [[Biot–Savart law]]
 
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), [http://www.ampere.cnrs.fr/ice/ice_page_detail.php?lang=fr&type=text&bdd=ampere&table=ampere_text&bookId=7&typeofbookDes=Textessci&pageOrder=71&facsimile=off&search=no free online version]</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.
 
==Ørsted's discovery==
The Danish physicist [[Hans Christian Oersted|Ørsted]] noticed in April 1820, while experimenting with the [[Alessandro Volta|Voltaic]] pile, an effect of an electric  current on a permanent magnet. He wrote a Latin publication which he sent round Europe. [[François Arago]] demonstrated the discovery at the French Académie des Sciences (11 September 1820), which inspired [[Jean-Baptiste Biot]] and [[André-Marie Ampère]] to investigate the effect further. At a meeting of the Académie des Sciences  on 30 October 1820,  Biot and Savart announced that the magnetic force exerted by a long conductor on a magnetic pole falls off with the reciprocal of the distance squared and is orientated perpendicular to the wire.  Simultaneously they published this in a short note. Laplace generalized their result mathematically.
 
==Laplace's formula==
{{Image|Laplace magnetic.png|right|250px|Magnetic induction d'''B''' at point '''r''' due to  infinitesimal piece d'''s''' (red) of wire (blue) transporting electric current ''i''. }}
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 [[SI|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 [[speed of light]]). 
 
If we remember the fact that the vector <math>\scriptstyle \vec{\mathbf{r}}</math> has dimension length, we see that this equation is an [[Inverse-square law]].
 
==Infinite straight conductor==
{{Image|Biot Savart.png|left|250px|Field '''B''' due to current ''i'' in infinitely long straight wire.}}
Take a straight, infinitely long,  conducting wire transporting the current ''i''.  Historically, this is the case considered by [[Biot]] and [[Savart]]. The magnetic field ''H'' at a distance ''R''
from the wire is (in SI units)
 
&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>
\qquad H = \frac{i}{ 2\pi R},
</math>
 
where in vacuum the [[magnetic field]] and the [[magnetic induction]] are related by '''B''' = &mu;<sub>0</sub>'''H'''.
In order to prove the expression for ''H'', we write, using  ''R'' = ''r''&thinsp;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 along <math>\scriptstyle \mathbf{B}</math>, which is  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> is moved 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}} = \hat{\mathbf{e}} \,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>
Substitution of ''y'' = ''s'' / R and ''y'' = cot&phi; = cos&phi; / sin&phi;, successively, gives
finally,
:<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'' is the distance of the point of observation to the wire (length of  vector <math>\scriptstyle \vec{\mathbf{R}}</math>).
 
The constant ''k'' depends on the choice of electromagnetic units and is &mu;<sub>0</sub>/4&pi; = 10<sup>&minus;7</sup> [[henry (unit)|henry]]/m  in rationalized [[SI]] units. In Gaussian units ''k'' = 1/''c''.  This equation gives the original formulation of Biot and Savart.
 
The SI dimension of ''B'' is T ([[tesla (unit)|tesla]]). A field of 1 T corresponds to 10&thinsp;000 G ([[gauss (unit)|gauss]]).
 
==Magnetic field on axis of circular current==
{{Image|Axial magnet.png|right|250px|A current ''i'' runs on circle of radius ''r'' in ''x''-''y'' plane. At a  distance ''R'' of the center there is a magnetic induction '''B''' along the ''z''-axis.}} The Biot-Savart expression can be integrated in [[cylinder coordinates]] for a point ''R'' on the ''z''-axis. This gives (see figure on the right) a  vector parallel to the ''z''-axis of magnitude
:<math>
B = 2\pi k \frac{i r^2}{(R^2+r^2)^{3/2}}.
</math>
Note that the direction of the electric current ''i'' and the magnetic induction '''B''' are related according to the right-hand screw rule: the screw is driven into the direction of '''B''' and rotated along ''i''. The factor ''k'' is &mu;<sub>0</sub>/4&pi; for SI units and 1/''c'' for Gaussian units.
 
==Generalized Biot-Savart-Laplace law==
:''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 a surface integral over the cross section ''A''. Rather than introducing a surface element, we multiply immediately by d''s'' and obtain an infinitesimal ''volume'' element <math>\scriptstyle  d\mathbf{r}' \equiv dV</math>,
{{Image|Piece of wire Biot Savart.png|left|150px|Piece of wire of cross section ''A'' and length <i>d</i>''s'', volume ''d''&thinsp;<b>r'</b> &equiv; ''dV'' = ''A''<i>d</i>''s'' }}
:<math>
i d\mathbf{s} \rightarrow \iint_{A} \mathbf{J}(\mathbf{r}') d\mathbf{r}',
</math> 
where we defined the current density '''J''' as a vector parallel to the line segment d'''s''' and of magnitude ''J''(<b>r</b>'). The volume element has height |d'''s'''| and  base an infinitesimal surface element of the cross section ''A''. The '''B'''-field at point '''r''', due to a volume ''V'' = ''As'' of the current 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, the [[gradient]], to act on the ''unprimed'' coordinates and hence it may be moved outside the integral, giving the following generalized form of the ''Biot-Savart-Laplace 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}', \qquad\qquad\qquad\qquad\qquad\qquad\qquad(1).
</math>
Here <math>\scriptstyle \boldsymbol{\nabla} \times </math> is the [[curl]] and  ''V'' is a finite volume of the current generating the '''B'''-field. The total '''B'''-field is obtained by having ''V'' encompass all current (integrating over all current).
 
==Consistency with Maxwell equations==
The expression for '''B''' given in Eq. (1) is a solution of the [[Maxwell equations]] with zero [[displacement current]]. This is of interest as it confirms the generally accepted notion that the Maxwell equations form a set of postulates for classical electrodynamics. All electrodynamic results, including Biot-Savart's law, must be derivable from them. We will show that the Biot-Savart-Laplace law can indeed be seen as a consequence of the Maxwell postulates, although Biot and Savart made their discovery some forty-five years before Maxwell.
 
The [[Helmholtz decomposition]] of '''B'''('''r''') is,
:<math>
\mathbf{B}(\mathbf{r}) = \mathbf{B}_\perp(\mathbf{r}) + \mathbf{B}_\parallel(\mathbf{r})
</math>
where
:<math>
\begin{align}
\mathbf{B}_\perp(\mathbf{r}) &= \frac{1}{4\pi}\boldsymbol{\nabla} \times
\iiint \frac{\boldsymbol{\nabla}'\times \mathbf{B}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
d\mathbf{r}'  \\
\mathbf{B}_\parallel(\mathbf{r}) &= -\frac{1}{4\pi}\boldsymbol{\nabla}
\iiint \frac{\boldsymbol{\nabla}'\cdot \mathbf{B}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
d\mathbf{r}' \\
\end{align}
</math>
are, respectively, the perpendicular (transverse, divergence-free) and parallel (longitudinal, curl-free)  components. The operator <b>&nabla;</b> acts on unprimed coordinates and <b>&nabla;'</b> acts on primed coordinates. The pertinent Maxwell's equations are in SI units
:<math>
\begin{align}
\boldsymbol{\nabla}'\cdot \mathbf{B}(\mathbf{r}') &= 0 \\
\boldsymbol{\nabla}'\times \mathbf{B}(\mathbf{r}') & = \mu_0 \mathbf{J}(\mathbf{r}')
\end{align}
</math>
Hence
:<math>
\mathbf{B}(\mathbf{r}) =k \boldsymbol{\nabla} \times
\iiint \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}
d\mathbf{r}', \qquad\hbox{with}\qquad  k= \frac{\mu_0 }{4\pi}.
</math>
 
==See also==
*[[Ampere's equation]]
*[[Ampere's law]]
 
==References==
<references />
Further reading:
* J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, John Wiley, New York (1975).

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