40-555: The ampere or amp (symbol A) is the base unit of electric current in the International System of Units. Ampere or Ampère may also refer to: Ampere The ampere ( / ˈ æ m p ɛər / AM -pair , US : / ˈ æ m p ɪər / AM -peer ; symbol: A ), often shortened to amp , is the unit of electric current in the International System of Units (SI). One ampere
80-489: A relative uncertainty of approximately a few parts in 10 , and involved realisations of the watt, the ohm and the volt. The 2019 revision of the SI defined the ampere by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 × 10 when expressed in the unit C, which is equal to A⋅s, where the second is defined in terms of ∆ ν Cs , the unperturbed ground state hyperfine transition frequency of
120-449: A silver nitrate solution. Later, more accurate measurements revealed that this current is 0.999 85 A . Since power is defined as the product of current and voltage, the ampere can alternatively be expressed in terms of the other units using the relationship I = P / V , and thus 1 A = 1 W/V. Current can be measured by a multimeter , a device that can measure electrical voltage, current, and resistance. Until 2019,
160-1424: 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 ′ d s d s ′ 1 r 2 [ ( ( ∂ r ∂ s ∂ r ∂ s ′ − 2 r ∂ 2 r ∂ s ∂ s ′ ) + r ∂ 2 Q ∂ s ∂ s ′ ) cos ( r x ) + ∂ Q ∂ s ′ cos ( x d s ) − ∂ Q ∂ s cos ( x d s ′ ) ] . {\displaystyle d^{2}F_{x}=kII'dsds'{\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].} Q
200-799: A number of scientists, including Wilhelm Weber , Rudolf Clausius , Maxwell, Bernhard Riemann , Hermann Grassmann , and Walther Ritz , developed this expression to find a fundamental expression of the force. Through differentiation, it can be shown that: cos ( x d s ) cos ( r d s ′ ) r 2 = − cos ( r x ) ( cos ε − 3 cos ϕ cos ϕ ′ ) r 2 . {\displaystyle {\frac {\cos(x\,ds)\cos(r\,ds')}{r^{2}}}=-\cos(rx){\frac {(\cos \varepsilon -3\cos \phi \cos \phi ')}{r^{2}}}.} and also
240-862: Is 1.2 A") and the charge accumulated (or passed through a circuit) over a period of time is expressed in coulombs (as in "the battery charge is 30 000 C "). The relation of the ampere (C/s) to the coulomb is the same as that of the watt (J/s) to the joule . The international system of units (SI) is based on seven SI base units the second , metre, kilogram , kelvin , ampere, mole , and candela representing seven fundamental types of physical quantity, or "dimensions" , ( time , length , mass , temperature , electric current, amount of substance , and luminous intensity respectively) with all other SI units being defined using these. These SI derived units can either be given special names e.g. watt, volt, lux , etc. or defined in terms of others, e.g. metre per second . The units with special names derived from
280-1821: 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 = − ( 1 + k ) 2 r {\displaystyle Q=-{\frac {(1+k)}{2r}}} We obtain the general expression for the force exerted on ds by ds : d 2 F = − k I I ′ 2 r 2 [ ( 3 − k ) r ^ 1 ( d s d s ′ ) − 3 ( 1 − k ) r ^ 1 ( r ^ 1 d s ) ( r ^ 1 d s ′ ) − ( 1 + k ) d s ( r ^ 1 d s ′ ) − ( 1 + k ) d s ′ ( r ^ 1 d s ) ] . {\displaystyle d^{2}\mathbf {F} =-{\frac {kII'}{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].} Integrating around s ' eliminates k and
320-413: Is along the x-axis, and wire 1 is at y=D, z=0, parallel to the x-axis. Let x 1 , x 2 {\displaystyle 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 ) {\displaystyle (x_{1},D,0)} and
360-722: 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 {\displaystyle L_{1}} . Then the force vector felt by wire 1 is: F 12 = μ 0 I 1 I 2 4 π 2 D ( 0 , − 1 , 0 ) L 1 . {\displaystyle \mathbf {F} _{12}={\frac {\mu _{0}I_{1}I_{2}}{4\pi }}{\frac {2}{D}}(0,-1,0)L_{1}.} As expected,
400-427: Is an electric current equivalent to 10 elementary charges moving every 1.602 176 634 seconds or 6.241 509 074 × 10 elementary charges moving in a second. Prior to the redefinition the ampere was defined as the current passing through two parallel wires 1 metre apart that produces a magnetic force of 2 × 10 newtons per metre. The earlier CGS system has two units of current, one structured similarly to
440-421: Is equal to 1 coulomb (C) moving past a point per second. It is named after French mathematician and physicist André-Marie Ampère (1775–1836), considered the father of electromagnetism along with Danish physicist Hans Christian Ørsted . As of the 2019 revision of the SI , the ampere is defined by fixing the elementary charge e to be exactly 1.602 176 634 × 10 C , which means an ampere
SECTION 10
#1732776877980480-411: 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 . The form of Ampere's force law commonly given was derived by James Clerk Maxwell in 1873 and is one of several expressions consistent with the original experiments of André-Marie Ampère and Carl Friedrich Gauss . The x -component of
520-548: Is the magnetic force constant from the Biot–Savart law , F m / L {\displaystyle 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 {\displaystyle r} is the distance between the two wires, and I 1 {\displaystyle I_{1}} , I 2 {\displaystyle I_{2}} are
560-406: Is used in the formal definition of the ampere. The SI unit of charge, the coulomb , was then defined as "the quantity of electricity carried in 1 second by a current of 1 ampere". Conversely, a current of one ampere is one coulomb of charge going past a given point per second: In general, charge Q was determined by steady current I flowing for a time t as Q = It . This definition of
600-446: The caesium -133 atom. The SI unit of charge, the coulomb , "is the quantity of electricity carried in 1 second by a current of 1 ampere". Conversely, a current of one ampere is one coulomb of charge going past a given point per second: In general, charge Q is determined by steady current I flowing for a time t as Q = I t . Constant, instantaneous and average current are expressed in amperes (as in "the charging current
640-505: 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 k A {\displaystyle k_{\rm {A}}} depends upon
680-952: The magnetic constant , in SI units 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. F 12 = μ 0 4 π ∫ L 1 ∫ L 2 I 1 d ℓ 1 × ( I 2 d ℓ 2 × r ^ 21 ) | r | 2 , {\displaystyle \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}}},} where To determine
720-1245: The Ampère expression: d 2 F = − k I I ′ r 3 [ 2 r ( d s d s ′ ) − 3 r ( r d s ) ( r d s ′ ) ] {\displaystyle d^{2}\mathbf {F} =-{\frac {kII'}{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]} If we take k=+1, we obtain d 2 F = − k I I ′ r 3 [ r ( d s d s ′ ) − d s ( r d s ′ ) − d s ′ ( r d s ) ] {\displaystyle d^{2}\mathbf {F} =-{\frac {kII'}{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]} Using
760-478: The SI defined the ampere as follows: The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2 × 10 newtons per metre of length. Ampère's force law states that there is an attractive or repulsive force between two parallel wires carrying an electric current. This force
800-508: The SI's and the other using Coulomb's law as a fundamental relationship, with the CGS unit of charge defined by measuring the force between two charged metal plates. The CGS unit of current is then defined as one unit of charge per second. The ampere is named for French physicist and mathematician André-Marie Ampère (1775–1836), who studied electromagnetism and laid the foundation of electrodynamics . In recognition of Ampère's contributions to
840-452: The amount of current that generates a force of two dynes per centimetre of length between two wires one centimetre apart. The size of the unit was chosen so that the units derived from it in the MKSA system would be conveniently sized. The "international ampere" was an early realization of the ampere, defined as the current that would deposit 0.001 118 grams of silver per second from
SECTION 20
#1732776877980880-415: The ampere are: There are also some SI units that are frequently used in the context of electrical engineering and electrical appliances, but are defined independently of the ampere, notably the hertz , joule, watt, candela, lumen , and lux. Like other SI units, the ampere can be modified by adding a prefix that multiplies it by a power of 10 . Amp%C3%A8re%27s force law In magnetostatics ,
920-485: The ampere was most accurately realised using a Kibble balance , but in practice the unit was maintained via Ohm's law from the units of electromotive force and resistance , the volt and the ohm , since the latter two could be tied to physical phenomena that are relatively easy to reproduce, the Josephson effect and the quantum Hall effect , respectively. Techniques to establish the realisation of an ampere had
960-410: The creation of modern electrical science, an international convention, signed at the 1881 International Exposition of Electricity , established the ampere as a standard unit of electrical measurement for electric current. The ampere was originally defined as one tenth of the unit of electric current in the centimetre–gram–second system of units . That unit, now known as the abampere , was defined as
1000-1289: The cross-product: F 12 = μ 0 I 1 I 2 4 π ∫ L 1 ∫ L 2 d x 1 d x 2 ( 0 , − D , 0 ) | ( x 1 − x 2 ) 2 + D 2 | 3 / 2 . {\displaystyle \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}}}.} Next, we integrate x 2 {\displaystyle x_{2}} from − ∞ {\displaystyle -\infty } to + ∞ {\displaystyle +\infty } : F 12 = μ 0 I 1 I 2 4 π 2 D ( 0 , − 1 , 0 ) ∫ L 1 d x 1 . {\displaystyle \mathbf {F} _{12}={\frac {\mu _{0}I_{1}I_{2}}{4\pi }}{\frac {2}{D}}(0,-1,0)\int _{L_{1}}dx_{1}.} If wire 1
1040-497: The definition of the ampere , the SI unit of electric current, states that the magnetic force per unit length between two straight parallel conductors is F m L = 2 k A I 1 I 2 r , {\displaystyle {\frac {F_{m}}{L}}=2k_{\rm {A}}{\frac {I_{1}I_{2}}{r}},} where k A {\displaystyle k_{\rm {A}}}
1080-1122: The differential element of wire 2 is at ( x 2 , 0 , 0 ) {\displaystyle (x_{2},0,0)} . By properties of line integrals, d ℓ 1 = ( d x 1 , 0 , 0 ) {\displaystyle d{\boldsymbol {\ell }}_{1}=(dx_{1},0,0)} and d ℓ 2 = ( d x 2 , 0 , 0 ) {\displaystyle d{\boldsymbol {\ell }}_{2}=(dx_{2},0,0)} . Also, r ^ 21 = 1 ( x 1 − x 2 ) 2 + D 2 ( x 1 − x 2 , D , 0 ) {\displaystyle {\hat {\mathbf {r} }}_{21}={\frac {1}{\sqrt {(x_{1}-x_{2})^{2}+D^{2}}}}(x_{1}-x_{2},D,0)} and | r | = ( x 1 − x 2 ) 2 + D 2 {\displaystyle |r|={\sqrt {(x_{1}-x_{2})^{2}+D^{2}}}} Therefore,
1120-661: 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: d F x = k I I ′ d s ′ ∫ d s cos ( x d s ) cos ( r d s ′ ) − cos ( r x ) cos ( d s d s ′ ) r 2 . {\displaystyle dF_{x}=kII'ds'\int ds{\frac {\cos(xds)\cos(rds')-\cos(rx)\cos(dsds')}{r^{2}}}.} Following Ampère,
1160-1003: 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: F 12 = − μ 0 4 π ∫ L 1 ∫ L 2 ( I 1 d ℓ 1 ⋅ I 2 d ℓ 2 ) r ^ 21 | r | 2 . {\displaystyle \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}}}.} In this form, it
1200-505: 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 . The best-known and simplest example of Ampère's force law, which underlaid (before 20 May 2019 )
1240-434: The force that the wire feels is proportional to its length. The force per unit length is: F 12 L 1 = μ 0 I 1 I 2 2 π D ( 0 , − 1 , 0 ) . {\displaystyle {\frac {\mathbf {F} _{12}}{L_{1}}}={\frac {\mu _{0}I_{1}I_{2}}{2\pi D}}(0,-1,0).} The direction of
Ampère - Misplaced Pages Continue
1280-684: The form: d 2 F = k I I ′ d s d s ′ r 2 ( ∂ r ∂ s ∂ r ∂ s ′ − 2 r ∂ 2 r ∂ s ∂ s ′ ) . {\displaystyle d^{2}F={\frac {kII'dsds'}{r^{2}}}\left({\frac {\partial r}{\partial s}}{\frac {\partial r}{\partial s'}}-2r{\frac {\partial ^{2}r}{\partial s\partial s'}}\right).} As Maxwell noted, terms can be added to this expression, which are derivatives of
1320-723: The general formula: F 12 = μ 0 4 π ∫ L 1 ∫ L 2 I 1 d ℓ 1 × ( I 2 d ℓ 2 × r ^ 21 ) | r | 2 , {\displaystyle \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}}},} Assume wire 2
1360-823: The identities: ∂ r ∂ s = cos ϕ , ∂ r ∂ s ′ = − cos ϕ ′ . {\displaystyle {\frac {\partial r}{\partial s}}=\cos \phi ,{\frac {\partial r}{\partial s'}}=-\cos \phi '.} and ∂ 2 r ∂ s ∂ s ′ = − cos ε + cos ϕ cos ϕ ′ r . {\displaystyle {\frac {\partial ^{2}r}{\partial s\partial s'}}={\frac {-\cos \varepsilon +\cos \phi \cos \phi '}{r}}.} Ampère's results can be expressed in
1400-924: The identity: cos ( r x ) cos ( d s d s ′ ) r 2 = cos ( r x ) cos ε r 2 . {\displaystyle {\frac {\cos(rx)\cos(ds\,ds')}{r^{2}}}={\frac {\cos(rx)\cos \varepsilon }{r^{2}}}.} With these expressions, Ampère's force law can be expressed as: d F x = k I I ′ d s ′ ∫ d s ′ cos ( r x ) 2 cos ε − 3 cos ϕ cos ϕ ′ r 2 . {\displaystyle dF_{x}=kII'ds'\int ds'\cos(rx){\frac {2\cos \varepsilon -3\cos \phi \cos \phi '}{r^{2}}}.} Using
1440-896: The integral is F 12 = μ 0 I 1 I 2 4 π ∫ L 1 ∫ L 2 ( d x 1 , 0 , 0 ) × [ ( d x 2 , 0 , 0 ) × ( x 1 − x 2 , D , 0 ) ] | ( x 1 − x 2 ) 2 + D 2 | 3 / 2 . {\displaystyle \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}}}.} Evaluating
1480-431: 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
1520-438: The second term is zero, and thus we find the form of Ampère's force law given by Maxwell: F = k I I ′ ∬ d s × ( d s ′ × r ) | r | 3 {\displaystyle \mathbf {F} =kII'\iint {\frac {d\mathbf {s} \times (d\mathbf {s} '\times \mathbf {r} )}{|r|^{3}}}} Start from
1560-518: The system of units chosen, and the value of k A {\displaystyle k_{\rm {A}}} decides how large the unit of current will be. In the SI system, k A = d e f μ 0 4 π {\displaystyle k_{\rm {A}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\mu _{0}}{4\pi }}} with μ 0 {\displaystyle \mu _{0}}
1600-589: The vector identity for the triple cross product, we may express this result as d 2 F = k I I ′ r 3 [ ( d s × d s ′ × r ) + d s ′ ( r d s ) ] {\displaystyle d^{2}\mathbf {F} ={\frac {kII'}{r^{3}}}\left[\left(d\mathbf {s} \times d\mathbf {s'} \times \mathbf {r} \right)+d\mathbf {s} '(\mathbf {r} \,d\mathbf {s} )\right]} When integrated around ds '
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