Ampère's law of force

The upper wire with current I ₁ experiences a Lorentz force F ₁₂ due to the magnetic field B ₂ that the lower wire generates. (The mirror image of the Lorentz force on the lower wire is not shown.)

According to Biot-Savart's law, there is a magnetic field around a current-carrying conductor and a Lorentz force on a second current-carrying conductor, i.e. two current-carrying conductors exert a force on each other. This relationship is also referred to in the literature as Ampère's law of force - not to be confused with Ampère's law (which is also known as the law of flow ).

Historical development

After Hans Christian Oersted showed at the beginning of 1820 that a current-carrying wire affects a compass needle, i.e. has a magnetic field, and in the same year Jean-Baptiste Biot and Félix Savart formulated the relationship between current flow and magnetic field (Biot-Savart law) , André discovered -Marie Ampère in the same year that a force occurs between parallel currents. He put his law on this in 1826 in his work Théorie mathématique des phénomènes électro-dynamiques uniquement déduite de l'expérience in differential form. Ampère's differential version differs from the differential version used today by Hermann Graßmann , no difference can be discovered in the experiment, because there are always closed circuits and in the integral version both formulations give the same result. In the following, the current Graßmann formulation is used, although the Ampère formulation has the advantage that it is compatible with the law of interaction in its differential form , which does not apply to Graßmann's formulation. The latter, however, has the advantage that it can now easily be derived from the Biot-Savart law and the Lorentz force .

Integral formula for two thin conductors

If one uses the differential formulation of Graßmann and integrates it, the result is for the force which is exerted by the current-carrying thin conductor 2 on the current-carrying thin conductor 1: ${\displaystyle {\vec {F}}_{12}}$

${\displaystyle {\vec {F}}_{12}={\frac {\mu _{0}}{4\pi }}I_{1}I_{2}\int _{L_{1}}\int _{L_{2}}{\frac {{\text{d}}{\vec {\ell }}_{1}\ \times \ ({\text{d}}{\vec {\ell }}_{2}\ \times \ {\vec {r}}_{21})}{r^{3}}}}$

in which

• ${\displaystyle \mu _{0}}$is the magnetic field constant ,
• ${\displaystyle I_{1}}$and the currents in conductor 1 and 2 are ${\displaystyle I_{2}}$
• ${\displaystyle {\text{d}}{\vec {\ell }}_{1}}$and the (infinitesimally small) vectorial line elements are at the location or (i.e. ) the two conductors, via which integration takes place in the double line integral along the curves and , ${\displaystyle {\text{d}}{\vec {\ell }}_{2}}$ ${\displaystyle {\vec {r}}_{1}}$ ${\displaystyle {\vec {r}}_{2}}$ ${\displaystyle {\text{d}}{\vec {\ell }}_{i}={\text{d}}{\vec {r}}_{i}}$ ${\displaystyle L_{1}}$ ${\displaystyle L_{2}}$
• ${\displaystyle {\vec {r}}_{21}={\vec {r}}_{1}-{\vec {r}}_{2}}$ is the vector pointing from the location of the line element of conductor 2 to the location of the line element of conductor 1,
• ${\displaystyle r=|{\vec {r}}_{21}|}$ is the distance between the two line elements,
• ${\displaystyle \times }$is the symbol for the cross product (vector product) and
• the currents or in conductor 1 or conductor 2 are constant. Their signs are to be considered relative to the orientation of or ; so if is, the line element always points in the direction of the technical current direction , which must be taken into account when parameterizing the curves . ${\displaystyle I_{1}}$ ${\displaystyle I_{2}}$ ${\displaystyle {\text{d}}{\vec {\ell }}_{1}}$ ${\displaystyle {\text{d}}{\vec {\ell }}_{2}}$ ${\displaystyle I_{1}>0}$ ${\displaystyle {\text{d}}{\vec {\ell }}_{1}}$ ${\displaystyle {\vec {r}}_{i}(t)}$

For the force exerted by the current flowing through the thin conductor 1 to the current-carrying thin conductor 2, applies to the interaction law : ${\displaystyle {\vec {F}}_{21}}$ ${\displaystyle {\vec {F}}_{21}=-{\vec {F}}_{12}}$

Requirements for the validity of the formula:

• The wires are uncharged.
• The current through the wires is constant.
• The wires are (ideally) thin and don't move.

Differential formulations

Differential formulation according to Graßmann:

${\displaystyle {\text{d}}^{2}{\vec {F}}_{12}^{(G)}={\frac {\mu _{0}}{4\pi }}I_{1}I_{2}{\frac {{\text{d}}{\vec {\ell }}_{1}\times ({\text{d}}{\vec {\ell }}_{2}\times {\vec {r}}_{21})}{r^{3}}}}$

Dissolving the double cross product with the Graßmann identity results in:

${\displaystyle {\text{d}}^{2}{\vec {F}}_{12}^{(G)}=-{\frac {\mu _{0}}{4\pi }}{\frac {I_{1}I_{2}}{r^{3}}}\left({\vec {r}}_{21}({\text{d}}{\vec {\ell }}_{1}\cdot {\text{d}}{\vec {\ell }}_{2})-{\text{d}}{\vec {\ell }}_{2}({\text{d}}{\vec {\ell }}_{1}\cdot {\vec {r}}_{21})\right)}$

Differential formulation according to Ampère:

${\displaystyle {\text{d}}^{2}{\vec {F}}_{12}^{(A)}=-{\frac {\mu _{0}}{4\pi }}{\frac {I_{1}I_{2}}{r^{3}}}{\vec {r}}_{21}\left(2{\text{d}}{\vec {\ell }}_{1}\cdot {\text{d}}{\vec {\ell }}_{2}-3{\frac {{\vec {r}}_{21}\cdot {\text{d}}{\vec {\ell }}_{1}}{r}}{\frac {{\vec {r}}_{21}\cdot {\text{d}}{\vec {\ell }}_{2}}{r}}\right)}$

where here is the symbol for the scalar product . ${\displaystyle \cdot }$

Now applies , because and the expression in brackets is symmetrical with the indices swapped. This means that the differential form is already compatible with the law of interaction , whereas this does not apply to. ${\displaystyle {\text{d}}^{2}{\vec {F}}_{21}^{(A)}=-{\text{d}}^{2}{\vec {F}}_{12}^{(A)}}$ ${\displaystyle {\vec {r}}_{21}=-{\vec {r}}_{12}}$ ${\displaystyle {\text{d}}^{2}{\vec {F}}_{12}^{(A)}}$ ${\displaystyle {\text{d}}^{2}{\vec {F}}_{12}^{(G)}}$

Linking Lorentzkraft and Biot-Savart

The following applies for the Lorentz force on the thin current-carrying conductor 1:

${\displaystyle {\vec {F}}_{L}=I_{1}\int _{L_{1}}\left({\text{d}}{\vec {\ell }}_{1}\times {\vec {B}}_{2}({\vec {r}}_{1})\right)}$Wherein the magnetic field of the current-carrying conductor 2 at the site is ${\displaystyle {\vec {B}}_{2}({\vec {r}}_{1})}$ ${\displaystyle {\vec {r}}_{1}}$

According to Biot-Savart's law, assuming that conductor 2 is thin:

${\displaystyle {\vec {B}}_{2}({\vec {r}}_{1})={\frac {\mu _{0}I_{2}}{4\pi }}\int _{L_{2}}{\frac {{\text{d}}{\vec {\ell }}_{2}\times {\vec {r}}_{21}}{r^{3}}}}$

If you put in the formula above, you get: ${\displaystyle {\vec {B}}_{2}({\vec {r}}_{1})}$

${\displaystyle {\vec {F}}_{L}=I_{1}\int _{L_{1}}\left({\text{d}}{\vec {\ell }}_{1}\times \left({\frac {\mu _{0}I_{2}}{4\pi }}\int _{L_{2}}{\frac {{\text{d}}{\vec {\ell }}_{2}\times {\vec {r}}_{21}}{r^{3}}}\right)\right)}$

And after extracting the scalar and constant factor it follows: ${\displaystyle {\frac {\mu _{0}I_{2}}{4\pi }}}$

${\displaystyle {\vec {F}}_{L}={\frac {\mu _{0}I_{1}I_{2}}{4\pi }}\int _{L_{1}}\left({\text{d}}{\vec {\ell }}_{1}\times \int _{L_{2}}{\frac {{\text{d}}{\vec {\ell }}_{2}\times {\vec {r}}_{21}}{r^{3}}}\right)}$

Since integral and cross product are linear operators, the following applies (assuming absolute integrability):

${\displaystyle {\vec {F}}_{L}={\frac {\mu _{0}I_{1}I_{2}}{4\pi }}\int _{L_{1}}\int _{L_{2}}{\frac {{\text{d}}{\vec {\ell }}_{1}\times ({\text{d}}{\vec {\ell }}_{2}\times {\vec {r}}_{21})}{r^{3}}}}$

Special case for parallel conductors

Illustration of the historical definition of the ampere

When the two conductors straight, thin, parallel and infinitely long, there is already from the valid to 2019 amperes Definition ago known simple formula for the amount of the successive forces or : ${\displaystyle F}$ ${\displaystyle {\vec {F}}_{12}}$ ${\displaystyle {\vec {F}}_{21}}$

${\displaystyle {\frac {F}{\ell }}={\frac {\mu _{0}}{2\pi }}\cdot {\frac {I_{1}\cdot I_{2}}{r}}=2\cdot 10^{-7}{\frac {\text{N}}{{\text{A}}^{2}}}\cdot {\frac {I_{1}\cdot I_{2}}{r}}}$

It is the force in relation to a conductor portion of the length , the distance between the conductor and or are the current levels in conductor 1 and 2 respectively. ${\displaystyle {\frac {F}{\ell }}}$ ${\displaystyle F}$ ${\displaystyle \ell }$ ${\displaystyle r}$ ${\displaystyle I_{1}}$ ${\displaystyle I_{2}}$

Web links

• Script for the Ampèreschen Kraftgesetz (website of the Humboldt-Universität zu Berlin )
• Ampère's Force Law ( University of Surrey website )
• Electromagnetic Theory: A Critical Examination of Fundamentals (formerly titled: Electromagnetics) (book by Alfred O'Rahilly, Dover (1965) on archive.org ) goes into great detail in Chapter IV on the different formulations.

Individual evidence

1. ↑ The German expression "Ampère'sches Kraftgesetz" occurs in current literature and teaching, see z. B. Electrodynamics (Dietmar Petrascheck, Franz Schwabl, Springer , 2nd edition (2014)) or Ampere's Law (script, see #Weblinks ), but comparatively seldom, because a Google search for the term yielded e.g. B. only 58 hits. The English counterpart “Ampère's force law”, on the other hand, is much more common, the expression provides over 2000 hits and since February 2008 has had its own Wikipedia article. Accessed on May 19, 2016.
2. ↑ This Month in Physics History - July 1820: Oersted and electromagnetism (article on www.aps.org, accessed May 21, 2016)
3. ↑ Article on Félix Savart (www-groups.dcs.st-and.ac.uk, accessed on May 21, 2016)
4. ↑ ^{a b} EVOLUTION OF ELECTROMAGNETICS IN THE 19TH CENTURY (article on www.ee.bgu.ac.il)
5. ↑ ^{a b c} Equivalence Between Ampère's and Grassmann's Forces (IEEE Transactions on Magnetics, Vol. 32, No. 2, March 1996)
6. ↑ The concept of the Lorentz force was only established in 1895, see Lexicon of Physics: Lorentz (www.spektrum.de)
7. ↑ With the integral, the wire is viewed as a curve (mathematics) .
8. ↑ see also BIPM SI Units brochure, 8th Edition, p. 105
9. ↑ see Biot-Savart Law (hyperphysics.phy-astr.gsu.edu, accessed on May 19, 2016)
10. ^ Tai L. Chow: Introduction to electromagnetic theory: a modern perspective . Jones and Bartlett, Boston 2006, ISBN 0-7637-3827-1 , p. 153.