Ampère's force law

1 Derivation of parallel straight wire case from general formula
2 References and notes
3 Further reading
4 See also

Electromagnetism

Electricity · Magnetism Electrostatics
Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential · Electrostatic induction · Electric dipole moment · Polarization density

Magnetostatics
Ampère’s law · Electric current · Magnetic field · Magnetization · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss's law for magnetism

Electrodynamics
Free space · Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy current

Electrical Network
Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides

Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Electromagnetic four-potential

Scientists
Ampère · Coulomb · Faraday · Gauss · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Volta · Weber · Ørsted

The top wire with current I₁ experiences a Lorentz force F₁₂ due to magnetic field B₂ created by the bottom wire. (Not shown is the simultaneous process where the bottom wire I₂ experiences a magnetic force F₂₁ due to magnetic field B₁ created by the top wire.

Another depiction of the Lorentz force law, showing both the force F₁₂ on wire 1 due to magnetic field of wire 2, and the equal and opposite force F₂₁ on wire 2 due to magnetic field of wire 1.

In magnetostatics, the force of attraction or repulsion between two current-carrying wires (see Figure 1) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, as defined by the Biot-Savart law, and the other wire experiences a magnetic force as a consequence, as defined by the Lorentz force.

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

$F_m = 2 k_A \frac {I_1 I_2 } {r}$ ,

where k_A is the magnetic force constant, r is the separation of the wires, and I₁, I₂ are the DC currents carried by the wires. This is a good approximation for finite lengths if the distance between the wires is small compared to their lengths. The value of k_A depends upon the system of units chosen, and the value of k_A decides how large the unit of current will be. In the SI system,

$k_A \ \overset{\underset{\mathrm{def}}{}}{=}\ \frac {\mu_0}{ 4 \pi} \$

with μ₀ the magnetic constant, defined in SI units as

$\mu_0 \ \overset{\underset{\mathrm{def}}{}}{=}\ 4 \pi \times 10^{-7} \$ newtons / (ampere)².

Thus, in vacuum, the force per meter of length between the two parallel conductors carrying a current of 1 A and spaced apart by 1 m, is exactly 2 × 10⁻⁷ N/m.

The general formulation of the magnetic force for arbitrary geometries is based on line integrals and combines the Biot-Savart law and Lorentz force in one equation as shown below. :

$\vec{F}_{12} = \frac {\mu_0} {4 \pi} \int_{L_1} \int_{L_2} \frac {I_1 d \vec{\ell}_1\ \mathbf{ \times} \ (I_2 d \vec{\ell}_2 \ \mathbf{ \times } \ \hat{\mathbf{r}}_{21} )} {|r|^2}$ ,

where

$\vec{F}_{12}$ is the total 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,
$d \vec{\ell}_1$ and $d \vec{\ell}_2$ are infinitesimal vectors associated with wire 1 and wire 2 respectively (usually measured in metres); see line integral for a detailed definition,
The vector $\hat{\mathbf{r}}_{21}$ 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 I_n is relative to the orientation $d \vec{\ell}_n$ (for example, if $d \vec{\ell}_1$ 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.

Derivation of parallel straight wire case from general formula

Start from the general formula:

$\vec{F}_{12} = \frac {\mu_0} {4 \pi} \int_{L_1} \int_{L_2} \frac {I_1 d \vec{\ell}_1\ \mathbf{ \times} \ (I_2 d \vec{\ell}_2 \ \mathbf{ \times } \ \hat{\mathbf{r}}_{21} )} {|r|^2}$ ,

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 $(x 2, D,0)$ . By properties of line integrals, $d\vec{\ell}_1=(dx_1,0,0)$ and $d\vec{\ell}_2=(dx_2,0,0)$ . Also,

$\hat{\mathbf{r}}_{21} = \frac{1}{\sqrt{(x_1-x_2)^2+D^2}}(x_1-x_2,D,0)$

and

$|r| = \sqrt{(x_1-x_2)^2+D^2}$

Therefore the integral is

$\vec{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi} \int_{L_1} \int_{L_2} \frac {(dx_1,0,0)\ \mathbf{ \times} \ ((dx_2,0,0) \ \mathbf{ \times } \ (x_1-x_2,D,0) )} {|(x_1-x_2)^2+D^2|^{3/2}}$ .

Evaluating the cross-product:

$\vec{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi} \int_{L_1} \int_{L_2} dx_1 dx_2 \frac {(0,-D,0)} {|(x_1-x_2)^2+D^2|^{3/2}}$ .

Next, we integrate $x 2$ from $-\infty$ to $+\infty$ :

$\vec{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi}\frac{2}{D}(0,-1,0) \int_{L_1} dx_1$ .

If wire 1 is also infinite, the integral diverges, because the total attractive force between two infinite parallel wires is infinity. In fact, we want to know 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:

$\vec{F}_{12} = \frac {\mu_0 I_1 I_2} {4 \pi}\frac{2}{D}(0,-1,0) L_1$ .

As expected, the force that the wire feels is proportional to its length. The force per unit length is:

$\frac{\vec{F}_{12}}{L_1} = \frac {\mu_0 I_1 I_2} {2 \pi D}(0,-1,0)$ .

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 $F m$ shown above.

References and notes

^ Raymond A Serway & Jewett JW (2006). Serway's principles of physics: a calculus based text (Fourth Edition ed.). Belmont, CA: Thompson Brooks/Cole. p. 746. ISBN 053449143X. http://books.google.com/books?id=1DZz341Pp50C&pg=RA1-PA746&dq=wire+%22magnetic+force%22&lr=&as_brr=0&sig=4vMV_CH6Nm8ZkgjtDJFlupekYoA#PRA1-PA746,M1.
^ Paul M. S. Monk (2004). Physical chemistry: understanding our chemical world. New York: Chichester: Wiley. p. 16. ISBN 0471491810. http://books.google.com/books?vid=ISBN0471491802&id=LupAi35QjhoC&pg=PA16&lpg=PA16&ots=IMiGyIL-67&dq=ampere+definition+si&sig=9Y0k0wgvymmLNYFMcXodwJZwvAM.
^ BIPM definition
^ "Magnetic constant". 2006 CODATA recommended values. NIST. http://physics.nist.gov/cgi-bin/cuu/Value?mu0. Retrieved 2007-08-08.
^ The integrand of this expression appears in the official documentation regarding definition of the ampere BIPM SI Units brochure, 8^th Edition, p. 105
^ Tai L. Chow (2006). Introduction to electromagnetic theory: a modern perspective. Boston: Jones and Bartlett. p. 153. ISBN 0763738271. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&lpg=PA153&dq=%22ampere's+law+of+force%22&source=web&ots=uZOFz9dWv7&sig=NJp3UQvbCOvcVm7eJN4IUdlC9bs.
^ Ampère's Force Law Scroll to section "Integral Equation" for formula.

Ampère's force law

Contents

Derivation of parallel straight wire case from general formula

References and notes

Further reading

See also