Biot-Savart law: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Paul Wormer
imported>John R. Brews
(Redirect)
 
(26 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{subpages}}
#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)</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 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 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|thumb|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 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 <math>\scriptstyle \vec{\mathbf{r}}</math> 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> 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
:<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).
 
==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>,
:<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]]. 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.
 
Since it is known from [[vector analysis]] that
:<math>
\boldsymbol{\nabla}\cdot\Big(\boldsymbol{\nabla}\times \mathbf{V}(\mathbf{r})\Big) = 0
</math>
for any vector field '''V'''('''r'''), it is immediately clear that '''B''' in Eq. (1) satisfies the following Maxwell equation:
:<math>
\boldsymbol{\nabla}\cdot\mathbf{B}(\mathbf{r}) = 0 .
</math>
This equation states in fact that electric current cannot give a magnetic charge (magnetic monopole).
The one other Maxwell equation of interest is
:<math>
\boldsymbol{\nabla}\times\mathbf{B}(\mathbf{r}) = 4\pi k \mathbf{J}(\mathbf{r}),
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(2)
</math>
where we assumed that there are no time-dependent electric fields present i.e., that  [[displacement current]]s are zero. This equation was first formulated in [[Ampere's law|integral form]] by [[André-Marie Ampère|Ampère]], as was acknowledged by [[James Clerk Maxwell|Maxwell]]. In SI units 4&pi;''k'' is equal to the vacuum permeability &mu;<sub>0</sub>.
In the proof we will need the following mathematical results
:<math>
\begin{align}
\boldsymbol{\nabla}\times( \boldsymbol{\nabla}\times \mathbf{V})  &= \boldsymbol{\nabla} (\boldsymbol{\nabla}\cdot\mathbf{V}) - \nabla^2 \mathbf{V} \\
\boldsymbol{\nabla} \frac{1}{|\mathbf{r}-\mathbf{r'}|} &= -\boldsymbol{\nabla}' \frac{1}{|\mathbf{r}-\mathbf{r'}|} \\
\nabla^2 \frac{1}{|\mathbf{r}-\mathbf{r'}|} &= - 4\pi \delta(\mathbf{r}-\mathbf{r'}) \\
\boldsymbol{\nabla} \cdot \mathbf{J} &= 0 .\\
\end{align}
</math>
The first equation is well-known in vector analysis, the second follows by differentiating to the components of '''r''' and <b>r'</b> and equating. The third equation has a [[Dirac delta function]] on the right-hand side and follows from [[distribution (mathematics)|distribution theory]]. The last equation follows from charge density (&rho;) conservation and the [[continuity equation]],
:<math>
\frac{\partial \rho}{\partial t} = 0 \quad \hbox{and}\quad  \boldsymbol{\nabla} \cdot \mathbf{J}+\frac{\partial \rho}{\partial t} = 0,
</math>
respectively. This continuity equation for electric charge can be derived from the Maxwell equations. Now, upon substitution of Eq. (1) into the left-hand side of Eq. (2), we obtain:
:<math>
\begin{align}
\boldsymbol{\nabla}\times \mathbf{B}(\mathbf{r}) &= k \boldsymbol{\nabla}\times  \boldsymbol{\nabla} \times \iiint_{V} \frac{\mathbf{J}(\mathbf{r}')}  {|\mathbf{r}-\mathbf{r'}|}d\mathbf{r}' \\
&= k \boldsymbol{\nabla}\iiint \mathbf{J}(\mathbf{r}')\cdot \boldsymbol{\nabla} \left( \frac{1} {|\mathbf{r}-\mathbf{r'}|}\right) d\mathbf{r}' - k  \iiint \mathbf{J}(\mathbf{r}') \nabla^2 \left( \frac{1}{|\mathbf{r}-\mathbf{r'}|}\right) d\mathbf{r}' \\
&= k \boldsymbol{\nabla}\iiint \Big(\boldsymbol{\nabla}' \cdot \mathbf{J}(\mathbf{r}')\Big) \left( \frac{1} {|\mathbf{r}-\mathbf{r'}|}\right) d\mathbf{r}' +  4\pi k \iiint \mathbf{J}(\mathbf{r}') \delta(\mathbf{r}-\mathbf{r'})  d\mathbf{r}' . \\
\end{align}
</math>
In the first term of the last step we replaced the unprimed nabla by a primed one (times minus) and then applied a turn-over rule with the primed nabla. This turn-over rule can be justified by partial integration and gives a minus sign, or, in other words, nabla is an [[anti-hermitian operator]]. Using that the divergence of the current density vanishes and the defining property of the delta function, we get finally
:<math>
\boldsymbol{\nabla}\times \mathbf{B}(\mathbf{r}) = 4\pi k \mathbf{J}(\mathbf{r}).
</math>
This shows that  '''B'''('''r''') of Eq. (1) (the generalized Biot-Savart-Laplace law) substituted into the left-hand side of the Maxwell equation (2) is equal to the right-hand side of this Maxwell equation  and hence we conclude that  Eq. (1) is a solution of Eq. (2).
 
==See also==
*[[Ampere's equation]]
*[[Ampere's law]]
 
==References==
<references />
Further reading:
* J. D. Jackson, ''Classical Electrodynamics'', 2nd edition, John Wiley, New York (1975).

Latest revision as of 12:18, 22 April 2011

Redirect to:

Retrieved from "https://citizendium.org/wiki/index.php?title=Biot-Savart_law&oldid=401960"