Ampere's equation

In physics, more particularly in electrodynamics, Ampère's equation describes the force between two infinitesimal elements of electric-current-carrying wires. The equation is named for the early nineteenth century French physicist and mathematician André-Marie Ampère.

Rather than giving Ampère's original infinitesimal equation, which is not without problems, we will describe two common cases obtained by integration: a system consisting of two straight wires and a system of two closed loops. Since the integrals over disputed terms in Ampère's infinitesimal equation vanish, the equations for these integrated systems are generally accepted and, moreover, are in full agreement with experiment.

Electromagnetic units

Equations will be given in two common systems of electromagnetic units (SI and Gaussian units) and to that end we define the constant k as follows,

$k={\begin{cases}{\displaystyle {\frac {\mu _{0}}{4\pi }}}&{\hbox{for SI units}}\\\\{\displaystyle {\frac {1}{c^{2}}}}&{\hbox{for Gaussian units}}.\end{cases}}$

Here μ₀ is the magnetic constant (also known as vacuum permeability). The quantity c is the speed of light in vacuum, in SI units a defined value denoted by c₀ = 299 792 458 m s⁻¹ (exactly).

Two straight, infinite, and parallel wires

(PD) Image: John R. Brews
B-field from current i₂ in wire 2 causes force F on wire 1.

Consider two wires, one carrying an electric current i₁, the other i₂. Both currents are constant in time; the wires are infinite, straight, and parallel, and a distance r apart.

The magnetic flux density B₂ due to the wire with current i₂ is directed in circles about wire 2, and at the distance r has magnitude (in SI units):

$B_{2}={\frac {\mu _{0}i_{2}}{2\pi r}}\ .$

The force F₁ on wire 1 due to this magnetic flux density is found regarding the current as a movement of charge. A wire carrying a current i₂ in time dt moves a charge q = i₂ dt. In this time the charge moves a distance dℓ = v dt (with v the speed of the electrons traveling down the wire), suggesting i₂ dℓ = q v. The Lorentz force upon the charge subject to magnetic flux density B perpendicular to the flow of charge is then radially directed with magnitude F = qvB = i₂B dℓ, which is the force on each element of length dℓ of the wire. Thus, the force per unit length upon wire 1 is:

$F_{1}={\frac {\mu _{0}i_{1}i_{2}}{2\pi r}}\,$

which is Ampère's law for the force per unit length l of wire. The force exerted by wire 1 on wire 2 is the same, so the subscript on the force is unnecessary. In general units, the force per unit length is:

$F=2k{\frac {i_{1}i_{2}}{r}}.\,$

The force F is attractive if the currents run in the same direction and repulsive if they flow in opposite direction.

Definition of the ampere

This force between straight, parallel wires is used to define the SI unit of current, the ampere, symbol A. Take two infinitely long wires in vacuum at a distance r = 1 m, consider the force that one meter of these wires exert on one another (l = 1 m) and let this force be F = 2⋅10⁻⁷ N (newton). Then for i₁ = i₂ the current strengths are by definition both equal to one ampere (1 A). In SI units this implies that k = 10⁻⁷ N/A² and hence that the magnetic constant is:

μ₀ = 4π k = 4π⋅10⁻⁷ N/A².

(PD) Image: John R. Brews
Illustration of interacting current elements i₁dl₁ and i₂dl₂ from two closed current-carrying loops

Two loops

Let $\scriptstyle i_{1}$ and $\scriptstyle i_{2}$ be electric currents constant in time. They run in separate loops (closed curves) C₁ and C₂, see figure on the right, where all quantities are defined. The total force between two loops is given by the double path integral over the loops

$\mathbf {F} _{12}=-ki_{1}i_{2}\oint _{C_{1}}\oint _{C_{2}}{\frac {(d{\boldsymbol {l}}_{1}\cdot d{\boldsymbol {l}}_{2})\,\mathbf {r} _{12}}{|\mathbf {r} _{12}|^{3}}}.\qquad \qquad \qquad (1)$

Here r₁₂ = r₁−r₂ is the vector locating element 1 from the location of element 2, pointing from 2 to 1.

Alternative expression

One often finds the following expression for the force between two electric-current-carrying loops:

$\mathbf {F} _{12}=ki_{1}i_{2}\oint _{C_{1}}\oint _{C_{2}}{\frac {d{\boldsymbol {l}}_{1}\times (d{\boldsymbol {l}}_{2}\,\times \mathbf {r} _{12})}{|\mathbf {r} _{12}|^{3}}},\qquad \qquad \ \ (2)$

instead of the simpler expression in Eq. (1). Here the multiplication signs (×) indicate vector products. The integrand [expression under the integral of Eq. (2)] follows from the Biot-Savart-Laplace expression for the magnetic induction B(r₁) due to a segment of the second loop. Insertion of B(r₁) into the Lorentz force that acts on the current in segment dl₁ gives the integrand of Eq. (2).

The labeling of the segments being arbitrary, one would expect the same force (in absolute value) when the labels 1 and 2 are interchanged, or in other words, one would expect Newton's third law $\scriptstyle \mathbf {F} _{12}=-\mathbf {F} _{21}$ (action is minus reaction) to hold. This is not the case, the integrand in equation (2) is non-symmetric under interchange of labels 1 and 2 and hence the integral also appears to be non-symmetric. However, after integration the expression becomes antisymmetric (changes sign under interchange of 1 and 2) and hence satisfies Newton's third law.

To see this we note that the force in Eq. (2) has in fact two contributions, as follows from a result well-known in vector analysis,

${\frac {d{\boldsymbol {l}}_{1}\times (d{\boldsymbol {l}}_{2}\,\times \mathbf {r} _{12})}{|\mathbf {r} _{12}|^{3}}}={\frac {(d{\boldsymbol {l}}_{1}\cdot \mathbf {r} _{12})\,d{\boldsymbol {l}}_{2}}{|\mathbf {r} _{12}|^{3}}}-{\frac {(d{\boldsymbol {l}}_{1}\cdot d{\boldsymbol {l}}_{2})\,\mathbf {r} _{12}}{|\mathbf {r} _{12}|^{3}}}.$

The second contribution gives an integral that is obviously equal to Eq. (1), which is manifestly antisymmetric under interchange of labels (the dot product does not change, the vector r₁₂ changes sign). We will show that the first contribution vanishes after integration over a closed curve.

Write

${\frac {\mathbf {r} _{12}}{|\mathbf {r} _{12}|^{3}}}=-{\boldsymbol {\nabla }}_{1}\left({\frac {1}{|\mathbf {r} _{12}|}}\right)\equiv -\mathbf {A} ,$

where A is a short hand notation. Applying Stokes' theorem, we find that the first contribution becomes

$-ki_{1}i_{2}\oint _{C_{2}}\left[\oint _{C_{1}}(d{\boldsymbol {l}}_{1}\cdot \mathbf {A} )\,\right]d{\boldsymbol {l}}_{2}=-ki_{1}i_{2}\oint _{C_{2}}\left[\iint _{S}({\boldsymbol {\nabla }}_{1}\times \mathbf {A} )\cdot d\mathbf {S} _{1}\,\right]d{\boldsymbol {l}}_{2}.$

It is well-known in vector analysis that

${\boldsymbol {\nabla }}_{1}\times {\boldsymbol {\nabla }}_{1}\Phi =0$

for any scalar function Φ and in particular for Φ ≡ 1/|r₁₂|. Hence the first contribution to the force of Eq. (2) vanishes.

History

Ampère's original force law is:

$d^{2}\mathbf {F_{12}} =-d^{2}\mathbf {F_{21}} =i_{1}i_{2}{\frac {\mu _{0}}{4\pi }}\ {\frac {\mathbf {{\hat {u}}_{12}} }{r_{12}^{2}}}\left(2\ (d{\boldsymbol {l_{1}\cdot }}d{\boldsymbol {l_{2}}})-3(\mathbf {{\hat {u}}_{12}\cdot } d{\boldsymbol {l_{1}}})(\mathbf {{\hat {u}}_{12}\cdot } d{\boldsymbol {l_{2}}})\right)\ ,$

with û₁₂ a unit vector pointing along the line joining element 2 to element 1 and r₁₂ the length of this line. The force element is second order because it is a product of two infinitesimals. This force law leads to the same force between closed current loops as the more commonly used Grassmann's law presented above:

$d^{2}\mathbf {F_{12}} =i_{1}i_{2}{\frac {\mu _{0}}{4\pi }}\ {\frac {1}{r_{12}^{2}}}\left(\ (d{\boldsymbol {l_{1}\cdot }}d{\boldsymbol {l_{2}}})\mathbf {{\hat {u}}_{12}} -(\mathbf {{\hat {u}}_{12}\cdot } d{\boldsymbol {l_{1}}})d{\boldsymbol {l_{2}}}\right)\ .$

Grassmann's law is readily derived from the Biot-Savart law, and is consistent with space-time symmetry. Unlike Ampère's law, Grassmann's law obviously is not symmetric under exchange of the indices 1 and 2, and violates Newton's law of action and reaction, but this behavior stems from not being based upon action-at-a-distance, but being based instead upon a force mediated by a field, which itself has physical properties to take into account. There exists some debate over the ultimate form of the force law between current elements.