Magnetic moment

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The magnetic moment of a magnet is a measure of its tendency to align with a magnetic field. Both the magnetic moment and magnetic field may be considered to be vectors having a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. The magnetic field produced by a magnet is proportional to its magnetic moment as well. For example, a loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments. More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object.

Units

In the International System of Units (SI), the dimension of magnetic dipole moment is Area×current, or L²I (see examples below). This is the basis on which the unit of magnetic dipole moment is defined. The SI unit of magnetic dipole moment has two equivalent representations:

1 m²·A = 1 J/T.

In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment in CGS:

(ESU CGS) 1 statA·cm² = 3.33564095×10^-14 (m²·A or J/T)

and (more frequently used)

(EMU CGS and Gaussian-CGS) 1 erg/G = 1 abA·cm² = 10^-3 (m²·A or J/T).

The ratio of these two non-equivalent CGS units (EMU/ESU) is equal exactly to the speed of light in free space, expressed in cm/s.

All formulas in this article are correct in SI units, but in other unit systems, the formulas may need to be changed. For example, in SI units, a loop of current with current I and area A has magnetic moment I×A (see below), but in Gaussian units the magnetic moment is I×A/c.

Two kinds of magnetic sources

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: (1) motion of electric charges, such as electric currents, and (2) the intrinsic magnetism of elementary particles, such as the electron.

Contributions due to the sources of the first kind can be calculated from knowing the distribution of all the electric currents (or, alternatively, of all the electric charges and their velocities) inside the system, by using the formulas below. On the other hand, the magnitude of each elementary particle's intrinsic magnetic moment is a fixed number, often measured experimentally to a great precision. For example, any electron's magnetic moment is measured to be −9.284764×10⁻²⁴ J/T.[1] The direction of the magnetic moment of any elementary particle is entirely determined by the direction of its spin (the minus in front of the value above indicates that any electron's magnetic moment is antiparallel to its spin).

The net magnetic moment of any system is a vector sum of contributions from one or both types of sources. For example, the magnetic moment of an atom of hydrogen-1 (the lightest hydrogen isotope, consisting of a proton and an electron) is a vector sum of the following contributions: (1) the intrinsic moment of the electron, (2) the orbital motion of the electron around the proton, (3) the intrinsic moment of the proton. Similarly, the magnetic moment of a bar magnet is the sum of the intrinsic and orbital magnetic moments of the unpaired electrons of the magnet's material.

Magnetism and angular momentum

There exists a close connection between angular momentum and magnetism, expressed on a macroscopic scale in the Einstein-de Haas effect, or "rotation by magnetization," and its inverse, the Barnett effect, or "magnetization by rotation."[2]

At the atomic and sub-atomic scales, this connection is expressed by the ratio of magnetic moment to angular momentum, the gyromagnetic ratio.

Examples of magnetic moments

The introduction to this article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style. (October 2009)

Magnetic moment of circular coils

Magnetic moment of a current-carrying loop depends on its area and current. For example, the magnitude of magnetic moment for a single-turn circular coil of radius 5 cm carrying 1 A of current is determined as:

$\pi\times (0.05 \ \mathrm{m})^2\times(1\;\mathrm{A})\approx 0.008\;\mathrm{A}\cdot\mathrm{m}^2=0.008\;\mathrm{J/T}\;.$

The vector of this moment is pointing perpendicular to the plane of the loop, in the direction of the magnetic field at the center of the loop. (See the right-hand rule.) Knowing this value of the loop's magnetic moment can be used to establish the following facts:

At distances R much larger than the radius of the loop r = 0.05 m, the magnetic field produced by this loop will drop off as:

$(10^{-7}\;\mathrm{T}\cdot\mathrm{m/A})\times 2\times\frac{0.008\;\mathrm{A}\cdot\mathrm{m}^2}{R^3}\approx \frac{1.6\times 10^{-9}\;\mathrm{T}\cdot\mathrm{m}^3}{R^3}$ (along the loop's axis) and $-(10^{-7}\;\mathrm{T}\cdot\mathrm{m/A})\times \frac{0.008\;\mathrm{A}\cdot\mathrm{m}^2}{R^3}\approx -\frac{0.8\times 10^{-9}\;\mathrm{T}\cdot\mathrm{m}^3}{R^3}$ (in the loop's plane). Minus indicates the field direction opposite to the axial case.

10 - 7 = μ 0 / 4π

, see magnetic field produced by a magnetic moment for more details.

In the Earth's magnetic field of 0.5 G (5×10⁻⁵ T) perpendicular to the loop's axis, the loop (as well as the Earth) will experience a torque in newton-meters of

$\tau\approx (0.008\;\mathrm{J/T})\times (5\times10^{-5}\;\mathrm{T})= 4\times10^{-7} \ \mathrm{N} \cdot \mathrm{m}$ .

This torque can be used to make an electric compass.

If such compass is allowed to orient its axis parallel to the Earth's field, the amount of energy in joules released from the compass-Earth system is given by:

$U\approx 0.008\;\mathrm{J/T}\times 5\times10^{-5}\;\mathrm{T}= 4\times10^{-7} \ \mathrm{J}$ .

This energy can be dissipated into heat to overcome friction in the compass suspension system.

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