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magnetostatics Magnetostatics is the study of magnetic fields in systems where the electric currents, currents are steady current, steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary ...
, Ampère's force law describes the force of attraction or repulsion between two current-carrying wires. The physical origin of this force is that each wire generates a
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
, following the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
, 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) the definition of the
ampere The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
, the SI unit of electric current, states that the magnetic force per unit length between two straight parallel conductors is \frac = 2 k_ \frac , where k_ is the magnetic force constant from the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
, F_m / L is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter), r is the distance between the two wires, and I_1, I_2 are the
direct current Direct current (DC) is one-directional electric current, flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor (material), conductor such as a wire, but can also flow throug ...
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 k_ depends upon the system of units chosen, and the value of k_ decides how large the unit of current will be. In the SI system, k_ \ \overset\ \frac which is in SI units . Here \mu_0 is the magnetic constant which in SI units is


General case

The general formulation of the magnetic force for arbitrary geometries is based on iterated
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
s and combines the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
and Lorentz force in one equation as shown below. \mathbf_ = \frac \int_ \int_ \frac , where *\mathbf_ is the total magnetic force felt by wire 1 due to wire 2 (usually measured in
newtons The newton (symbol: N) is the unit of force in the International System of Units (SI). Expressed in terms of SI base units, it is 1 kg⋅m/s2, the force that accelerates a mass of one kilogram at one metre per second squared. The unit i ...
), *I_1 and I_2 are the currents running through wires 1 and 2, respectively (usually measured in
ampere The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
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, *d \boldsymbol_1 and d \boldsymbol_2 are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
s); see
line integral In mathematics, a line integral is an integral where the function (mathematics), function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integr ...
for a detailed definition, *The vector \hat_ is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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
vector cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
, *The sign of I_n is relative to the orientation d \boldsymbol_n (for example, if d \boldsymbol_1 points in the direction of conventional current, then I_1 > 0). To determine the force between wires in a material medium, the magnetic constant is replaced by the actual 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 and applying Stokes' theorem: \mathbf_ = -\frac \int_ \int_ \frac . 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 Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body r ...
.


Historical background

The form of Ampere's force law commonly given was derived by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism an ...
in 1873 and is one of several expressions consistent with the original experiments of André-Marie Ampère and
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
. 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: dF_x = k I I' ds' \int ds \frac . Following Ampère, a number of scientists, including Wilhelm Weber,
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
, Maxwell,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
,
Hermann Grassmann Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was littl ...
, and Walther Ritz, developed this expression to find a fundamental expression of the force. Through differentiation, it can be shown that: \frac = -\cos(rx) \frac . and also the identity: \frac = \frac . With these expressions, Ampère's force law can be expressed as: dF_x = k I I' ds' \int ds' \cos(rx) \frac . Using the identities: \frac = \cos\phi, \frac = -\cos\phi' . and \frac = \frac . Ampère's results can be expressed in the form: d^2 F = \frac \left( \frac \frac - 2r \frac\right). 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''': d^2 F_x = k I I' ds ds'\frac \left \left( \left( \frac \frac - 2r \frac\right) + r \frac\right) \cos(rx) + \frac \cos(x\,ds) - \frac \cos(x\,ds') \right. ''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: Q = - \frac We obtain the general expression for the force exerted on ''ds'' by ''ds : d^2\mathbf = -\frac \left[ \left(3-k\right) \hat_1 \left(d\mathbf\,d\mathbf'\right) - 3\left(1-k\right) \hat_1 \left(\mathbf_1 d\mathbf\right) \left(\mathbf_1 d\mathbf'\right) - \left(1+k\right) d\mathbf \left(\mathbf_1 d\mathbf'\right) - \left(1+k\right) d\mathbf' \left(\mathbf_1 d\mathbf\right) \right] . Integrating around ''s''' 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: d^2\mathbf = -\frac \left 2 \mathbf (d\mathbf \, d\mathbf) - 3\mathbf (\mathbf d\mathbf) (\mathbf d\mathbf) \right If we take k=+1, we obtain d^2\mathbf = -\frac \left \mathbf \left(d\mathbf \, d\mathbf\right) - d\mathbf \left(\mathbf \, d\mathbf'\right) - d\mathbf'\left(\mathbf \, d\mathbf\right) \right Using the vector identity for the triple cross product, we may express this result as d^2\mathbf = \frac \left \left(d\mathbf \times d\mathbf\times\mathbf\right) + d\mathbf'(\mathbf \, d\mathbf) \right When integrated around ''ds''' the second term is zero, and thus we find the form of Ampère's force law given by Maxwell: \mathbf = k I I' \iint \frac


Derivation of parallel straight wire case from general formula

Start from the general formula: \mathbf_ = \frac \int_ \int_ \frac , Assume wire 2 is along the x-axis, and wire 1 is at y=D, z=0, parallel to the x-axis. Let x_1,x_2 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 (x_1,D,0) and the differential element of wire 2 is at (x_2,0,0). By properties of line integrals, d\boldsymbol_1=(dx_1,0,0) and d\boldsymbol_2=(dx_2,0,0). Also, \hat_ = \frac(x_1-x_2,D,0) and , r, = \sqrt Therefore, the integral is \mathbf_ = \frac \int_ \int_ \frac . Evaluating the cross-product: \mathbf_ = \frac \int_ \int_ dx_1 dx_2 \frac . Next, we integrate x_2 from -\infty to +\infty: \mathbf_ = \frac \frac(0,-1,0) \int_ dx_1 . 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 L_1. Then the force vector felt by wire 1 is: \mathbf_ = \frac \frac(0,-1,0) L_1 . As expected, the force that the wire feels is proportional to its length. The force per unit length is: \frac = \frac (0,-1,0) . 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 \frac shown above.


Notable derivations

Chronologically ordered: *Ampère's original 1823 derivation: ** *
Maxwell Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of N ...
's 1873 derivation:
''Treatise on Electricity and Magnetism'' vol. 2, part 4, ch. 2 (§§502–527)
*
Pierre Duhem Pierre Maurice Marie Duhem (; 9 June 1861 – 14 September 1916) was a French theoretical physicist who made significant contributions to thermodynamics, hydrodynamics, and the theory of Elasticity (physics), elasticity. Duhem was also a prolif ...
's 1892 derivation: **
EPUB
***translation of
''Leçons sur l'électricité et le magnétisme'' vol. 3, appendix to book 14, pp. 309-332
* Alfred O'Rahilly's 1938 derivation:
''Electromagnetic Theory: A Critical Examination of Fundamentals'' vol. 1, pp. 102
��104 (cf. the following pages, too)


See also

*
Ampere The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
* Magnetic constant * Lorentz force * Ampère's circuital law *
Free space A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...


References and notes


External links


Ampère's force law
Includes animated graphic of the force vectors. {{DEFAULTSORT:Ampere's Force Law Ampere's law Ampere's law