View source for Physics:Ampère's force law

{{Short description|Physical law}}
{{Electromagnetism|cTopic=[[Physics:Magnetostatics|Magnetostatics]]}}

[[File:MagneticWireAttraction.svg|thumb|Two current-carrying wires attract each other magnetically: The bottom wire has current ''I''<sub>1</sub>, which creates magnetic field ''B''<sub>1</sub>. The top wire carries a current ''I''<sub>2</sub> through the magnetic field ''B''<sub>1</sub>, so (by the Lorentz force) the wire experiences a force ''F''<sub>12</sub>. (Not shown is the simultaneous process where the top wire makes a magnetic field which results in a force on the bottom wire.)]]

In [[Physics:Magnetostatics|magnetostatics]], the force of attraction or repulsion between two [[Physics:Electric current|current]]-carrying wires (see first figure below) is often called '''Ampère's force law'''. The physical origin of this force is that each wire generates a [[Physics:Magnetic field|magnetic field]], following the [[Physics:Biot–Savart law|Biot–Savart law]], and the other wire experiences a magnetic force as a consequence, following the Lorentz force law.

==Equation==

===Special case: Two straight parallel wires===

The best-known and simplest example of Ampère's force law, which underlaid (before 20 May 2019<ref>{{Cite web|title=26th CGPM Resolutions|url=https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf|access-date=2020-08-01|website=BIPM}}</ref>) the definition of the [[Ampere|ampere]], the SI unit of electric current, states that the magnetic force per unit length between two straight parallel conductors is 

<math display="block"> \frac {F_m} {L} = 2 k_{\rm A} \frac {I_1 I_2 } {r},</math> 

where <math>k_{\rm A}</math> is the magnetic force constant from the [[Physics:Biot–Savart law|Biot–Savart law]], <math>F_m / L</math> is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter), <math>r</math> is the distance between the two wires, and <math>I_1</math>, <math>I_2</math> are the [[Physics:Direct current|direct current]]s carried by the wires.

This is a good approximation if one wire is sufficiently longer than the other, so that it can be approximated as infinitely long, and if the distance between the wires is small compared to their lengths (so that the one infinite-wire approximation holds), but large compared to their diameters (so that they may also be approximated as infinitely thin lines). The value of <math>k_{\rm A}</math> depends upon the system of units chosen, and the value of <math>k_{\rm A}</math> decides how large the unit of current will be.

In the SI system,<ref name=Serway>{{cite book | author=Raymond A Serway & Jewett JW | title=Serway's principles of physics: a calculus based text | url=https://books.google.com/books?id=1DZz341Pp50C&q=wire+%22magnetic+force%22&pg=RA1-PA746 |publisher=Thompson Brooks/Cole | edition=Fourth | location=Belmont, California | year=2006 | page=746 | isbn=0-534-49143-X}}</ref><ref name=Monk>{{cite book | author=Paul M. S. Monk | title=Physical chemistry: understanding our chemical world | url=https://books.google.com/books?id=LupAi35QjhoC&q=ampere+definition+si&pg=PA16 |publisher=Chichester: Wiley | location=New York | year=2004 | page=16 | isbn=0-471-49181-0}}</ref>
<math display="block"> k_{\rm A} \ \overset{\underset{\mathrm{def}}{}}{=}\ \frac {\mu_0}{ 4\pi} </math>
with <math>\mu_0</math> the magnetic constant, in SI units
{{block indent|1=''μ''<sub>0</sub> = {{val|1.25663706212|(19)|e=-6|u=[[Henry (unit)|H]]/m}} }}

===General case===
The general formulation of the magnetic force for arbitrary geometries is based on iterated [[Line integral|line integral]]s and combines the [[Physics:Biot–Savart law|Biot–Savart law]] and [[Physics:Lorentz force|Lorentz force]] in one equation as shown below.<ref>The integrand of this expression appears in the official documentation regarding definition of the ampere [http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf BIPM SI Units brochure, 8th Edition, p. 105]</ref><ref name=Chow>{{cite book | author=Tai L. Chow | title=Introduction to electromagnetic theory: a modern perspective | url=https://books.google.com/books?id=dpnpMhw1zo8C&q=%22ampere's+law+of+force%22&pg=PA153 |publisher=Jones and Bartlett | location=Boston | year=2006  | page=153 | isbn=0-7637-3827-1}}</ref><ref>[http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/ampereForce.html Ampère's Force Law] '' Scroll to section "Integral Equation" for formula.''</ref>

<math display="block"> \mathbf{F}_{12} = \frac {\mu_0} {4 \pi} \int_{L_1} \int_{L_2} \frac {I_1 d \boldsymbol{\ell}_1\ \times \ (I_2 d \boldsymbol{\ell}_2 \ \times \ \hat{\mathbf{r}}_{21} )} {|r|^2},</math>

where
*<math>\mathbf{F}_{12}</math> is the total magnetic force felt by wire 1 due to wire 2 (usually measured in [[Newton (unit)|newtons]]),
*<math>I_1</math> and <math>I_2</math> are the currents running through wires 1 and 2, respectively (usually measured in [[Ampere|ampere]]s),
*The double line integration sums the force upon each element of wire 1 due to the magnetic field of each element of wire 2,
*<math>d \boldsymbol{\ell}_1</math> and <math>d \boldsymbol{\ell}_2</math> are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in [[Metre|metre]]s); see [[Line integral|line integral]] for a detailed definition,
*The vector <math>\hat{\mathbf{r}}_{21}</math> is the [[Unit vector|unit vector]] pointing from the differential element on wire 2 towards the differential element on wire 1, and ''|r|'' is the distance separating these elements,
*The multiplication '''×''' is a [[Cross product|vector cross product]],
*The sign of <math>I_n</math> is relative to the orientation <math>d \boldsymbol{\ell}_n</math> (for example, if <math>d \boldsymbol{\ell}_1</math> points in the direction of [[Physics:Conventional current|conventional current]], then <math>I_1 > 0</math>).

To determine the force between wires in a material medium, the magnetic constant is replaced by the actual [[Physics:Permeability (electromagnetism)|permeability]] of the medium.

For the case of two separate closed wires, the law can be rewritten in the following equivalent way by expanding the [[Vector triple product|vector triple product]] and applying Stokes' theorem:<ref>{{Cite journal| doi = 10.1119/1.15613| volume = 56| issue = 4| pages = 357–362| last = Christodoulides| first = C.| title = Comparison of the Ampère and Biot–Savart magnetostatic force laws in their line-current-element forms| journal = American Journal of Physics| year = 1988| url = https://www.researchgate.net/publication/243490933|bibcode = 1988AmJPh..56..357C }}</ref>
<math display="block"> \mathbf{F}_{12} = -\frac {\mu_0} {4 \pi} \int_{L_1} \int_{L_2} \frac {( I_1 d \boldsymbol{\ell}_1\ \mathbf{ \cdot} \ I_2 d \boldsymbol{\ell}_2 ) \ \hat{\mathbf{r}}_{21} } {|r|^2}.</math>

In this form, it is immediately obvious that the force on wire 1 due to wire 2 is equal and opposite the force on wire 2 due to wire 1, in accordance with Newton's third law of motion.

==Historical background==

[[Image:AmpereDiagram.jpg|thumbnail|500px|Diagram of original Ampere experiment]]

The form of Ampere's force law commonly given was derived by [[Biography:James Clerk Maxwell|James Clerk Maxwell]] in 1873 and is one of several expressions consistent with the original experiments of [[Biography:André-Marie Ampère|André-Marie Ampère]] and [[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]]. 
The ''x''-component of the force between two linear currents ''I'' and ''I{{’}}'', as depicted in the adjacent diagram, was given by Ampère in 1825 and Gauss in 1833 as follows:<ref>{{cite book|last=O'Rahilly|first=Alfred|title=Electromagnetic Theory|year=1965|publisher=Dover|pages=104|url=https://archive.org/stream/ElectrodynamicsORahilly#page/n113/mode/2up}} (cf. {{Cite journal| doi = 10.1051/jphystap:01886005002601| volume = 5| issue = 1| pages = 26–29| last = Duhem| first = P.|  title = Sur la loi d'Ampère| journal = J. Phys. Theor. Appl.| access-date = 2015-01-07| date = 1886| url = https://hal.archives-ouvertes.fr/jpa-00238641}}, which appears in {{Cite book| publisher = Gauthier-Villars| last = Duhem| first = Pierre Maurice Marie|author-link=Pierre Duhem| title = Leçons sur l'électricité et le magnétisme|volume=3| url=https://archive.org/stream/leonssurllectri01duhegoog#page/n320/mode/2up| location = Paris| date = 1891}})</ref>

<math display="block"> dF_x = k I I' ds' \int ds \frac {\cos(xds) \cos(rds') - \cos(rx)\cos(dsds')} {r^2}. </math>

Following Ampère, a number of scientists, including [[Biography:Wilhelm Eduard Weber|Wilhelm Weber]], [[Biography:Rudolf Clausius|Rudolf Clausius]], Maxwell, [[Biography:Bernhard Riemann|Bernhard Riemann]], [[Biography:Hermann Grassmann|Hermann Grassmann]],<ref>{{cite book|title=Hermann Graßmann : biography|last=Petsche|first=Hans-Joachim|publisher=Birkhäuser|year=2009|isbn=9783764388591|location=Basel Boston|pages=39}}</ref> and Walther Ritz, developed this expression to find a fundamental expression of the force. Through differentiation, it can be shown that:

<math display="block"> \frac{\cos(x\,ds) \cos(r\,ds')} {r^2} = -\cos(rx) \frac{(\cos\varepsilon - 3 \cos\phi \cos\phi')} {r^2}. </math>

and also the identity:
<math display="block"> \frac{\cos(rx) \cos(ds\,ds')} {r^2} = \frac{\cos(rx) \cos\varepsilon} {r^2} .</math>

With these expressions, Ampère's force law can be expressed as:
<math display="block"> dF_x = k I I' ds' \int ds' \cos(rx) \frac{2\cos\varepsilon - 3\cos\phi \cos\phi'} {r^2} .</math>

Using the identities:
<math display="block"> \frac{\partial r} {\partial s} = \cos\phi, \frac{\partial r} {\partial s'} = -\cos\phi' .</math>
and
<math display="block"> \frac{\partial^2 r} {\partial s \partial s'} = \frac{-\cos\varepsilon + \cos\phi \cos\phi'} {r} .</math>

Ampère's results can be expressed in the form:
<math display="block"> d^2 F = \frac{k I I' ds ds'} {r^2} \left( \frac{\partial r}{\partial s} \frac{\partial r}{\partial s'} - 2r \frac{\partial^2 r}{\partial s \partial s'}\right). </math>

As Maxwell noted, terms can be added to this expression, which are derivatives of a function ''Q''(''r'') and, when integrated, cancel each other out. Thus, Maxwell gave "the most general form consistent with the experimental facts" for the force on ''ds'' arising from the action of ''ds''<nowiki/>':<ref>{{cite book|last=Maxwell|first=James Clerk|title=Treatise on Electricity and Magnetism|year=1904| publisher=Oxford|pages=173}}</ref>
<math display="block"> d^2 F_x = k I I' ds ds'\frac{1} {r^2} \left[ \left( \left( \frac{\partial r}{\partial s} \frac{\partial r}{\partial s'} - 2r \frac{\partial^2 r}{\partial s \partial s'}\right) + r \frac{\partial^2 Q}{\partial s \partial s'}\right) \cos(rx) + \frac{\partial Q} {\partial s'} \cos(x\,ds) - \frac{\partial Q} {\partial s} \cos(x\,ds') \right] . </math>

''Q'' is a function of ''r'', according to Maxwell, which "cannot be determined, without assumptions of some kind, from experiments in which the active current forms a closed circuit." Taking the function ''Q''(''r'') to be of the form:
<math display="block"> Q = - \frac{(1+k)} {2r} </math>

We obtain the general expression for the force exerted on ''ds'' by ''ds'':
<math display="block"> d^2\mathbf{F} = -\frac{k I I'} {2r^2} \left[ \left(3-k\right) \hat{\mathbf{r}}_1 \left(d\mathbf{s}\,d\mathbf{s}'\right) - 3\left(1-k\right) \hat{\mathbf{r}}_1 \left(\mathbf{\hat{r}}_1 d\mathbf{s}\right) \left(\mathbf{\hat{r}}_1 d\mathbf{s}'\right) - \left(1+k\right) d\mathbf{s} \left(\mathbf{\hat{r}}_1 d\mathbf{s}'\right) - \left(1+k\right) d\mathbf{s}' \left(\mathbf{\hat{r}}_1 d\mathbf{s}\right) \right] .</math>

Integrating around ''s''<nowiki/>' eliminates ''k'' and the original expression given by Ampère and Gauss is obtained. Thus, as far as the original Ampère experiments are concerned, the value of k has no significance. Ampère took ''k''=−1; Gauss took ''k''=+1, as did Grassmann and Clausius, although Clausius omitted the ''S'' component. In the non-ethereal electron theories, Weber took ''k''=−1 and Riemann took ''k''=+1. Ritz left ''k'' undetermined in his theory. If we take ''k'' = −1, we obtain the Ampère expression:
<math display="block"> d^2\mathbf{F} = -\frac{k I I'} {r^3} \left[ 2 \mathbf{r} (d\mathbf{s} \, d\mathbf{s'}) - 3\mathbf{r} (\mathbf{r} d\mathbf{s}) (\mathbf{r} d\mathbf{s'}) \right] </math>

If we take k=+1, we obtain
<math display="block"> d^2\mathbf{F} = -\frac{k I I'} {r^3} \left[ \mathbf{r} \left(d\mathbf{s} \, d\mathbf{s'}\right) - d\mathbf{s} \left(\mathbf{r} \, d\mathbf{s}'\right) - d\mathbf{s}'\left(\mathbf{r} \, d\mathbf{s}\right) \right] </math>

Using the vector identity for the triple cross product, we may express this result as
<math display="block"> d^2\mathbf{F} = \frac{k I I'} {r^3} \left[ \left(d\mathbf{s} \times d\mathbf{s'}\times\mathbf{r}\right) + d\mathbf{s}'(\mathbf{r} \, d\mathbf{s}) \right] </math>

When integrated around ''ds''<nowiki/>' the second term is zero, and thus we find the form of Ampère's force law given by Maxwell:
<math display="block"> \mathbf{F} = k I I' \iint \frac{d\mathbf{s} \times (d\mathbf{s}' \times \mathbf{r})} {|r|^3} </math>

==Derivation of parallel straight wire case from general formula==

Start from the general formula:
<math display="block"> \mathbf{F}_{12} = \frac {\mu_0} {4 \pi} \int_{L_1} \int_{L_2} \frac {I_1 d \boldsymbol{\ell}_1\ \times \ (I_2 d \boldsymbol{\ell}_2 \ \times \ \hat{\mathbf{r}}_{21} )} {|r|^2},</math>
Assume wire 2 is along the x-axis, and wire 1 is at y=D, z=0, parallel to the x-axis. Let <math>x_1,x_2</math> be the ''x''-coordinate of the differential element of wire 1 and wire 2, respectively. In other words, the differential element of wire 1 is at <math>(x_1,D,0)</math> and the differential element of wire 2 is at <math>(x_2,0,0)</math>. By properties of line integrals, <math>d\boldsymbol{\ell}_1=(dx_1,0,0)</math> and <math>d\boldsymbol{\ell}_2=(dx_2,0,0)</math>. Also,
<math display="block"> \hat{\mathbf{r}}_{21} = \frac{1}{\sqrt{(x_1-x_2)^2+D^2}}(x_1-x_2,D,0)</math>
and <math display="block"> |r| = \sqrt{(x_1-x_2)^2+D^2}</math>
Therefore, the integral is
<math display="block"> \mathbf{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi} \int_{L_1} \int_{L_2} \frac {(dx_1,0,0)\ \times \ \left[ (dx_2,0,0) \ \times \ (x_1-x_2,D,0) \right ] } {|(x_1-x_2)^2+D^2|^{3/2}}.</math>
Evaluating the cross-product:
<math display="block"> \mathbf{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi} \int_{L_1} \int_{L_2} dx_1 dx_2 \frac {(0,-D,0)} {|(x_1-x_2)^2+D^2|^{3/2}}.</math>
Next, we integrate <math>x_2</math> from <math>-\infty</math> to <math>+\infty</math>:
<math display="block"> \mathbf{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi}\frac{2}{D}(0,-1,0) \int_{L_1} dx_1 .</math>
If wire 1 is also infinite, the integral diverges, because the ''total'' attractive force between two infinite parallel wires is infinity. In fact, what we really want to know is the attractive force ''per unit length'' of wire 1. Therefore, assume wire 1 has a large but finite length <math>L_1</math>. Then the force vector felt by wire 1 is:
<math display="block"> \mathbf{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi}\frac{2}{D}(0,-1,0) L_1 .</math>
As expected, the force that the wire feels is proportional to its length. The force per unit length is:
<math display="block"> \frac{\mathbf{F}_{12}}{L_1} = \frac {\mu_0 I_1 I_2} {2 \pi D}(0,-1,0) .</math>
The direction of the force is along the y-axis, representing wire 1 getting pulled towards wire 2 if the currents are parallel, as expected. The magnitude of the force per unit length agrees with the expression for <math>\frac {F_m} {L}</math> shown above.

==Notable derivations==
Chronologically ordered:
*Ampère's original 1823 derivation:
**{{Cite book| publisher = Apeiron| isbn = 978-1-987980-03-5| last1 = Assis| first1 = André Koch Torres| last2 = Chaib| first2 = J. P. M. C| last3 = Ampère| first3 = André-Marie| title = Ampère's electrodynamics: analysis of the meaning and evolution of Ampère's force between current elements, together with a complete translation of his masterpiece: Theory of electrodynamic phenomena, uniquely deduced from experience| location = Montreal| date = 2015| url=http://www.ifi.unicamp.br/~assis/Amperes-Electrodynamics.pdf}}
*[[Biography:James Clerk Maxwell|Maxwell]]'s 1873 derivation:
**[https://archive.org/stream/electricandmag02maxwrich#page/n177/mode/2up ''Treatise on Electricity and Magnetism'' vol. 2, part 4, ch. 2 (§§502–527)]
*[[Biography:Pierre Duhem|Pierre Duhem]]'s 1892 derivation:
** {{Cite book| last = Duhem| first = Pierre Maurice Marie| others = Alan Aversa (trans.)| title = Ampère's Force Law: A Modern Introduction| access-date = 2019-07-03| date = 2018-09-09| url = https://isidore.co/calibre/get/PDF/6855| doi=10.13140/RG.2.2.31100.03206/1}} ([https://isidore.co/calibre/get/EPUB/6855 EPUB])
***translation of: [https://archive.org/stream/leonssurllectri01duhegoog#page/n320/mode/2up ''Leçons sur l'électricité et le magnétisme'' vol. 3, appendix to book 14, pp. 309-332] {{in lang|fr}}
*Alfred O'Rahilly's 1938 derivation:
**[https://archive.org/stream/ElectrodynamicsORahilly#page/n111/mode/2up ''Electromagnetic Theory: A Critical Examination of Fundamentals'' vol. 1, pp. 102]–104 (cf. the following pages, too)

==See also==
* [[Ampere]]
* Magnetic constant
* [[Physics:Lorentz force|Lorentz force]]
* [[Physics:Ampère's circuital law|Ampère's circuital law]]
* [[Physics:Free space|Free space]]

==References and notes==
{{reflist}}

==External links==
*[http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/ampereForce.html Ampère's force law] Includes animated graphic of the force vectors.

{{DEFAULTSORT:Ampere's Force Law}}
[[Category:Electrostatics|Ampere's law]]
[[Category:Magnetostatics|Ampere's law]]

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