Magnetic dipole 1

Magnetic dipole

Dipole moment m.

Electric current I.The magnetic field and magnetic moment, due to an electric current or natural magnetic dipoles, either generatesthe same field profile.A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of thesource are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electricdipole, but the analogy is not complete. In particular, a magnetic monopole, the magnetic analogue of an electriccharge, has never been observed. Moreover, one form of magnetic dipole moment is associated with a fundamentalquantum property—the spin of elementary particles.The magnetic field around any magnetic source looks increasingly like the field of a magnetic dipole as the distancefrom the source increases.

External magnetic field produced by a magnetic dipole moment

An electrostatic analogue for a magneticmoment: two opposing charges separated

by a finite distance. Each arrowrepresents the direction of the field vector

at that point.

In classical physics, the magnetic field of a dipole is calculated as the limit ofeither a current loop or a pair of charges as the source shrinks to a point whilekeeping the magnetic moment m constant. For the current loop, this limit ismost easily derived for the vector potential. Outside of the source region, thispotential is (in SI units)

Magnetic dipole 2

The magnetic field of a current loop. The ringrepresents the current loop, which goes into the page at

the x and comes out at the dot.

with 4π r2 being the surface of a sphere of radius r;and the magnetic flux density (strength of the B-field) in teslas is

Alternatively one can obtain the scalar potential first from the magnetic pole limit,

and hence the magnetic field strength (or strength of the H-field) in ampere-turns per meter is

The magnetic field is symmetric under rotations about the axis of the magnetic moment.

Internal magnetic field of a dipoleThe two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field farfrom the source. However, inside the source region they give different predictions. The magnetic field between polesis in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge),while inside a current loop it is in the same direction (see the figure to the right). Clearly, the limits of these fieldsmust also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used tocalculate fields inside a magnetic material.If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current andarea constant, the limiting field is

.

where n=x/|x| is a unit vector, and δ(x) is the Dirac delta function in three dimensions. Unlike the expressions in theprevious section, this limit is correct for the internal field of the dipole.

Magnetic dipole 3

If a magnetic dipole is formed by taking a "north pole" and a "south pole", bringing them closer and closer togetherbut keeping the product of magnetic pole-charge and distance constant, the limiting field is

These fields are related by B = μ0(H+M), where

is the magnetization.

Forces between two magnetic dipolesThe force F exerted by one dipole moment m1 on another m2 separated in space by a vector r can be calculatedusing:

or

where r is the distance between dipoles. The force acting on m1 is in the opposite direction.The torque can be obtained from the formula

Dipolar fields from finite sourcesThe magnetic scalar potential ψ produced by a finite source, but external to it, can be represented by a multipoleexpansion. Each term in the expansion is associated with a characteristic moment and a potential having acharacteristic rate of decrease with distance r from the source. Monopole moments have a 1/r rate of decrease,dipole moments have a 1/r2 rate, quadrupole moments have a 1/r3 rate, and so on. The higher the order, the fasterthe potential drops off. Since the lowest-order term observed in magnetic sources is the dipolar term, it dominates atlarge distances. Therefore, at large distances any magnetic source looks like a dipole with the same magneticmoment.

Notes

References• Chow, Tai L. (2006). Introduction to electromagnetic theory: a modern perspective. Jones & Bartlett Learning.

ISBN 978-0-7637-3827-3.• Jackson, John D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X. OCLC  224523909

(http:/ / www. worldcat. org/ oclc/ 224523909).• Furlani, Edward P. (2001). Permanent Magnet and Electromechanical Devices: Materials, Analysis, and

Applications (http:/ / books. google. com/ ?id=irsdLnC5SrsC& dq=permanent+ magnet+ and+electromechanical+ devices& printsec=frontcover& q=3. 130). Academic Press. ISBN 0-12-269951-3.

• Schill, R. A. (2003). "General relation for the vector magnetic field of a circular current loop: A closer look".IEEE Transactions on Magnetics 39 (2): 961–967. Bibcode: 2003ITM....39..961S (http:/ / adsabs. harvard. edu/abs/ 2003ITM. . . . 39. . 961S). doi: 10.1109/TMAG.2003.808597 (http:/ / dx. doi. org/ 10. 1109/ TMAG. 2003.808597).

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