Ampère's force law

In magnetostatics, 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, and the other wire experiences a magnetic force as a consequence, following the Lorentz force law.

Two current-carrying wires attract each other magnetically: The bottom wire has current I1, which creates magnetic field B1. The top wire carries a current I2 through the magnetic field B1, so (by the Lorentz force) the wire experiences a force F12. (Not shown is the simultaneous process where the top wire makes a magnetic field which results in a force on the bottom wire.)


Special case: Two straight parallel wires

The best-known and simplest example of Ampère's force law, which underlies the definition of the ampere, the SI unit of current, states that the force per unit length between two straight parallel conductors is


where kA is the magnetic force constant from the Biot–Savart law, Fm/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 I1, I2 are the direct currents 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 kA depends upon the system of units chosen, and the value of kA decides how large the unit of current will be. In the SI system,[1] [2]

with μ0 the magnetic constant, defined in SI units as[3][4]

N / A2.

Thus, in vacuum,

the force per meter of length between two parallel conductors – spaced apart by 1 m and each carrying a current of 1 A – is exactly
N / m.

General case

The general formulation of the magnetic force for arbitrary geometries is based on iterated line integrals and combines the Biot–Savart law and Lorentz force in one equation as shown below.[5][6][7]



  • is the total magnetic force felt by wire 1 due to wire 2 (usually measured in newtons),
  • I1 and I2 are the currents running through wires 1 and 2, respectively (usually measured in amperes),
  • 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 metres); see line integral for a detailed definition,
  • The vector is the 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 vector cross product,
  • The sign of In is relative to the orientation (for example, if points in the direction of conventional current, then I1>0).

To determine the force between wires in a material medium, the magnetic constant is replaced by the actual permeability of the medium.

By expanding the vector triple product and applying Stokes' theorem, the law can be rewritten in the following equivalent way:[8]

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

Diagram of original Ampere experiment

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. 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:[9]

Following Ampère, a number of scientists, including Wilhelm Weber, Rudolf Clausius, James Clerk Maxwell, Bernhard Riemann, Hermann Grassmann,[10] and Walther Ritz, 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:




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':[11]


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,


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:

See also

References and notes

  1. ^ Raymond A Serway & Jewett JW (2006). Serway's principles of physics: a calculus based text (Fourth ed.). Belmont, California: Thompson Brooks/Cole. p. 746. ISBN 0-534-49143-X.
  2. ^ Paul M. S. Monk (2004). Physical chemistry: understanding our chemical world. New York: Chichester: Wiley. p. 16. ISBN 0-471-49181-0.
  3. ^ BIPM definition
  4. ^ "Magnetic constant". 2006 CODATA recommended values. NIST. Archived from the original on 20 August 2007. Retrieved 8 August 2007.
  5. ^ The integrand of this expression appears in the official documentation regarding definition of the ampere BIPM SI Units brochure, 8th Edition, p. 105
  6. ^ Tai L. Chow (2006). Introduction to electromagnetic theory: a modern perspective. Boston: Jones and Bartlett. p. 153. ISBN 0-7637-3827-1.
  7. ^ Ampère's Force Law Scroll to section "Integral Equation" for formula.
  8. ^ Christodoulides, C. (1988). "Comparison of the Ampère and Biot–Savart magnetostatic force laws in their line-current-element forms" (PDF). American Journal of Physics. 56 (4): 357–362. Bibcode:1988AmJPh..56..357C. doi:10.1119/1.15613.
  9. ^ O'Rahilly, Alfred (1965). Electromagnetic Theory. Dover. p. 104. (cf. Duhem, P. (1886). "Sur la loi d'Ampère". J. Phys. Theor. Appl. 5 (1): 26–29. doi:10.1051/jphystap:01886005002601. Retrieved 2015-01-07., which appears in Duhem, Pierre Maurice Marie (1891). Leçons sur l'électricité et le magnétisme. 3. Paris: Gauthier-Villars.)
  10. ^ Petsche, Hans-Joachim (2009). Hermann Graßmann : biography. Basel Boston: Birkhäuser. p. 39. ISBN 9783764388591.
  11. ^ Maxwell, James Clerk (1904). Treatise on Electricity and Magnetism. Oxford. p. 173.

External links


The ampere (; symbol: A), often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.

The International System of Units defines the ampere in terms of other base units by measuring the electromagnetic force between electrical conductors carrying electric current. The earlier CGS measurement system had two different definitions of current, one essentially the same as the SI's and the other using electric charge as the base unit, with the unit of charge defined by measuring the force between two charged metal plates. The ampere was then defined as one coulomb of charge per second. In SI, the unit of charge, the coulomb, is defined as the charge carried by one ampere during one second.

New definitions, in terms of invariant constants of nature, specifically the elementary charge, will take effect on 20 May 2019.

Ampère's circuital law

In classical electromagnetism, Ampère's circuital law (not to be confused with Ampère's force law that André-Marie Ampère discovered in 1823) relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell (not Ampère) derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

André-Marie Ampère

André-Marie Ampère (; French: [ɑ̃pɛʁ]; 20 January 1775 – 10 June 1836) was a French physicist and mathematician who was one of the founders of the science of classical electromagnetism, which he referred to as "electrodynamics". He is also the inventor of numerous applications, such as the solenoid (a term coined by him) and the electrical telegraph. An autodidact, Ampère was a member of the French Academy of Sciences and professor at the École polytechnique and the Collège de France.

The SI unit of measurement of electric current, the ampere, is named after him. His name is also one of the 72 names inscribed on the Eiffel Tower.

Birkeland current

A Birkeland current is a set of currents that flow along geomagnetic field lines connecting the Earth’s magnetosphere to the Earth's high latitude ionosphere. In the Earth’s magnetosphere, the currents are driven by the solar wind and interplanetary magnetic field and by bulk motions of plasma through the magnetosphere (convection indirectly driven by the interplanetary environment). The strength of the Birkeland currents changes with activity in the magnetosphere (e.g. during substorms). Small scale variations in the upward current sheets (downward flowing electrons) accelerate magnetospheric electrons which, when they reach the upper atmosphere, create the Auroras Borealis and Australis. In the high latitude ionosphere (or auroral zones), the Birkeland currents close through the region of the auroral electrojet, which flows perpendicular to the local magnetic field in the ionosphere. The Birkeland currents occur in two pairs of field-aligned current sheets. One pair extends from noon through the dusk sector to the midnight sector. The other pair extends from noon through the dawn sector to the midnight sector. The sheet on the high latitude side of the auroral zone is referred to as the Region 1 current sheet and the sheet on the low latitude side is referred to as the Region 2 current sheet.

The currents were predicted in 1908 by Norwegian explorer and physicist Kristian Birkeland, who undertook expeditions north of the Arctic Circle to study the aurora. He rediscovered, using simple magnetic field measurement instruments, that when the aurora appeared the needles of magnetometers changed direction, confirming the findings of Anders Celsius and assistant Olof Hjorter more than a century before. This could only imply that currents were flowing in the atmosphere above. He theorized that somehow the Sun emitted a cathode ray, and corpuscles from what is now known as a solar wind entered the Earth’s magnetic field and created currents, thereby creating the aurora. This view was scorned by other researchers, but in 1967 a satellite, launched into the auroral region, showed that the currents posited by Birkeland existed. In honour of him and his theory these currents are named Birkeland currents. A good description of the discoveries by Birkeland is given in the book by Jago.Professor Emeritus of the Alfvén Laboratory in Sweden, Carl-Gunne Fälthammar wrote: "A reason why Birkeland currents are particularly interesting is that, in the plasma forced to carry them, they cause a number of plasma physical processes to occur (waves, instabilities, fine structure formation). These in turn lead to consequences such as acceleration of charged particles, both positive and negative, and element separation (such as preferential ejection of oxygen ions). Both of these classes of phenomena should have a general astrophysical interest far beyond that of understanding the space environment of our own Earth."

Centimetre–gram–second system of units

The centimetre–gram–second system of units (abbreviated CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism.The CGS system has been largely supplanted by the MKS system based on the metre, kilogram, and second, which was in turn extended and replaced by the International System of Units (SI). In many fields of science and engineering, SI is the only system of units in use but there remain certain subfields where CGS is prevalent.

In measurements of purely mechanical systems (involving units of length, mass, force, energy, pressure, and so on), the differences between CGS and SI are straightforward and rather trivial; the unit-conversion factors are all powers of 10 as 100 cm = 1 m and 1000 g = 1 kg. For example, the CGS unit of force is the dyne which is defined as 1 g⋅cm/s2, so the SI unit of force, the newton (1 kg⋅m/s2), is equal to 100,000 dynes.

On the other hand, in measurements of electromagnetic phenomena (involving units of charge, electric and magnetic fields, voltage, and so on), converting between CGS and SI is more subtle. Formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on which system of units one uses. This is because there is no one-to-one correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "sub-systems", including Gaussian units, "ESU", "EMU", and Lorentz–Heaviside units. Among these choices, Gaussian units are the most common today, and "CGS units" often used specifically refers to CGS-Gaussian units.

Charles Babbage

Charles Babbage (; 26 December 1791 – 18 October 1871) was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage originated the concept of a digital programmable computer.Considered by some to be a "father of the computer", Babbage is credited with inventing the first mechanical computer that eventually led to more complex electronic designs, though all the essential ideas of modern computers are to be found in Babbage's analytical engine. His varied work in other fields has led him to be described as "pre-eminent" among the many polymaths of his century.Parts of Babbage's incomplete mechanisms are on display in the Science Museum in London. In 1991, a functioning difference engine was constructed from Babbage's original plans. Built to tolerances achievable in the 19th century, the success of the finished engine indicated that Babbage's machine would have worked.


The coulomb (symbol: C) is the International System of Units (SI) unit of electric charge. It is the charge (symbol: Q or q) transported by a constant current of one ampere in one second:

Thus, it is also the amount of excess charge on a capacitor of one farad charged to a potential difference of one volt:

The coulomb is equivalent to the charge of approximately 6.242×1018 (1.036×10−5 mol) protons, and −1 C is equivalent to the charge of approximately 6.242×1018 electrons.

A new definition, in terms of the elementary charge, will take effect on 20 May 2019. The new definition, defines the elementary charge (the charge of the proton) as exactly 1.602176634×10−19 coulombs. This would implicitly define the coulomb as ​10.1602176634×1018 elementary charges.

Electric current

An electric current is a flow of electric charge. In electric circuits this charge is often carried by moving electrons in a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionised gas (plasma).The SI unit for measuring an electric current is the ampere, which is the flow of electric charge across a surface at the rate of one coulomb per second. Electric current is measured using a device called an ammeter.Electric currents cause Joule heating, which creates light in incandescent light bulbs. They also create magnetic fields, which are used in motors, inductors and generators.

The moving charged particles in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom, and can move freely about within the metal. These conduction electrons are the charge carriers in metal conductors.

Electric motor

An electric motor is an electrical machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and winding currents to generate force in the form of rotation. Electric motors can be powered by direct current (DC) sources, such as from batteries, motor vehicles or rectifiers, or by alternating current (AC) sources, such as a power grid, inverters or electrical generators. An electric generator is mechanically identical to an electric motor, but operates in the reverse direction, accepting mechanical energy (such as from flowing water) and converting this mechanical energy into electrical energy.

Electric motors may be classified by considerations such as power source type, internal construction, application and type of motion output. In addition to AC versus DC types, motors may be brushed or brushless, may be of various phase (see single-phase, two-phase, or three-phase), and may be either air-cooled or liquid-cooled. General-purpose motors with standard dimensions and characteristics provide convenient mechanical power for industrial use. The largest electric motors are used for ship propulsion, pipeline compression and pumped-storage applications with ratings reaching 100 megawatts. Electric motors are found in industrial fans, blowers and pumps, machine tools, household appliances, power tools and disk drives. Small motors may be found in electric watches.

In certain applications, such as in regenerative braking with traction motors, electric motors can be used in reverse as generators to recover energy that might otherwise be lost as heat and friction.

Electric motors produce linear or rotary force (torque) and can be distinguished from devices such as magnetic solenoids and loudspeakers that convert electricity into motion but do not generate usable mechanical force, which are respectively referred to as actuators and transducers.

Glossary of electrical and electronics engineering

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Index of electrical engineering articles

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To navigate by individual letter use the table of contents below.

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 q moving with a velocity v in an electric field E and a magnetic field B experiences a force

(in SI units). Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a charged particle which might be traveling near the speed of light (relativistic form of the Lorentz force).

The first derivation of the Lorentz force is commonly attributed to Oliver Heaviside in 1889, although other historians suggest an earlier origin in an 1865 paper by James Clerk Maxwell. Hendrik Lorentz derived it in 1895, a few years after Heaviside.

Magnetic field

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials (such as iron) and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges (electric currents) such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.

The term 'magnetic field' is used for two distinct but closely related fields denoted by the symbols B and H. In the International System of Units, H is measured in units of amperes per meter and B is measured in teslas, which are equivalent to newtons per meter per ampere. H and B differ in how they account for magnetization. In a vacuum, B and H are the same aside from units; but in a magnetized material, B/ and H differ by the magnetization M of the material at that point in the material.

Magnetic fields are produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. Magnetic fields and electric fields are interrelated, and are both components of the electromagnetic force, one of the four fundamental forces of nature.

Magnetic fields are widely used throughout modern technology, particularly in electrical engineering and electromechanics. Rotating magnetic fields are used in both electric motors and generators. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits. Magnetic forces give information about the charge carriers in a material through the Hall effect. The Earth produces its own magnetic field, which shields the Earth's ozone layer from the solar wind and is important in navigation using a compass.


StarTram is a proposal for a maglev space launch system. The initial Generation 1 facility would be cargo only, launching from a mountain peak at an altitude of 3 to 7 kilometres (1.9 to 4.3 mi) with an evacuated tube staying at local surface level; it has been claimed that about 150,000 tons could be lifted to orbit annually. More advanced technology would be required for the Generation 2 system for passengers, with a longer track instead gradually curving up at its end to the thinner air at 22 kilometres (14 mi) altitude, supported by magnetic levitation, reducing g-forces when each capsule transitions from the vacuum tube to the atmosphere. A SPESIF 2010 presentation stated that Gen-1 could be completed by the year 2020+ if funding began in 2010, Gen-2 by 2030+.

Vacuum permittivity

The physical constant ε0 (pronounced as “epsilon nought” or “epsilon zero”), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value that can be approximated as

ε0 = 8.854187817...×10−12 F⋅m−1 (farads per metre).

It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

The value of the constant fraction is approximately 9 × 109 N⋅m2⋅C−2, q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

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