![MagneticWireAttraction](https://upload.wikimedia.org/wikipedia/commons/b/bc/MagneticWireAttraction.svg)
In
magnetostatics
Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the electric charge, charges are stationary. The magnetization need not be st ...
, the force of attraction or repulsion between two 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 magnetic field, 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
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
.
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 elect ...
, the
SI unit of current, states that the magnetic force per unit length between two straight parallel conductors is
where
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 ...
,
is the total force on either wire per unit length of the shorter (the longer is approximated as infinitely long relative to the shorter),
is the distance between the two wires, and
,
are the
direct current
Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
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
depends upon the system of units chosen, and the value of
decides how large the unit of current will be.
In the
SI system,
with
the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
, in SI units
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 to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
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 physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
in one equation as shown below.
where
*
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). It is defined as 1 kgâ‹…m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in r ...
),
*
and
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 elect ...
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,
*
and
are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
s); see
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
for a detailed definition,
*The vector
is the
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
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 is ...
,
*The sign of
is relative to the orientation
(for example, if
points in the direction of
conventional current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...
, then
).
To determine the force between wires in a material medium, the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical 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
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
and applying Stokes' theorem:
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 3rd law.
Historical background
The form of Ampere's force law commonly given was derived by
Maxwell and is one of several expressions consistent with the original experiments of
Ampère and
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
.
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:
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 ...
,
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
,
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
,
Hermann Grassmann
Hermann Günther Grassmann (german: link=no, 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 mat ...
, and
Walther Ritz
Walther Heinrich Wilhelm Ritz (22 February 1878 – 7 July 1909) was a Swiss theoretical physicist. He is most famous for his work with Johannes Rydberg on the Rydberg–Ritz combination principle. Ritz is also known for the variational method na ...
, developed this expression to find a fundamental expression of the force. Through differentiation, it can be shown that:
and also the identity:
With these expressions, Ampère's force law can be expressed as:
Using the identities:
and
Ampère's results can be expressed in the form:
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':
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:
We obtain the general expression for the force exerted on ds by ds:
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:
If we take k=+1, we obtain
Using the vector identity for the triple cross product, we may express this result as
When integrated around ds' the second term is zero, and thus we find the form of Ampère's force law given by Maxwell:
Derivation of parallel straight wire case from general formula
Start from the general formula:
Assume wire 2 is along the x-axis, and wire 1 is at y=D, z=0, parallel to the x-axis. Let
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
and the differential element of wire 2 is at
. By properties of line integrals,
and
. Also,
and
Therefore, the integral is
Evaluating the cross-product:
Next, we integrate
from
to
:
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
. Then the force vector felt by wire 1 is:
As expected, the force that the wire feels is proportional to its length. The force per unit length is:
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
shown above.
Notable derivations of Ampère's force law
Chronologically ordered:
*Ampère's original 1823 derivation:
**
*
Maxwell'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 worked on thermodynamics, hydrodynamics, and the theory of elasticity. Duhem was also a historian of science, noted for his work on the Euro ...
'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
Alfred O'Rahilly, KSG (1 October 1884 – 1 August 1969) was an academic with controversial views on both electromagnetism and religion. He briefly served in politics, as a Teachta Dála (TD) for Cork City, and was later the president of Unive ...
'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 elect ...
*
Magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
*
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
*
Ampère's circuital law
*
Free space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
References and notes
External links
Ampère's force lawIncludes animated graphic of the force vectors.
{{DEFAULTSORT:Ampere's Force Law
Ampere's law
Ampere's law