Inelectromagnetism, amagnetic dipole is the limit of either a closed loop ofelectric current or a pair of poles as the size of the source is reduced to zero while keeping themagnetic moment constant.

It is a magnetic analogue of theelectric dipole, but the analogy is not perfect. In particular, a truemagnetic monopole, the magnetic analogue of anelectric charge, has never been observed in nature. However, magnetic monopolequasiparticles have been observed as emergent properties of certain condensed matter systems.^[2] Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—thespin ofelementary particles.

Because magnetic monopoles do not exist, the magnetic field at a large distance from any static magnetic source looks like the field of a dipole with the same dipole moment. For higher-order sources (e.g.quadrupoles) with no dipole moment, their field decays towards zero with distance faster than a dipole field does.

Inclassical physics, the magnetic field of a dipole is calculated as the limit of either a current loop or a pair of charges as the source shrinks to a point while keeping themagnetic momentmconstant. For the current loop, this limit is most easily derived from thevector potential:^[3]

${\displaystyle \mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi r^2}\frac{\mathbf{m}\times \mathbf{r}}{r}=\frac{{\mu }_0}{4\pi }\frac{\mathbf{m}\times \mathbf{r}}{r^3},}$

whereμ₀ is thevacuum permeability constant and4π r² is the surface of a sphere of radiusr.The magnetic flux density (strength of the B-field) is then^[3]

${\displaystyle \mathbf{B}(\mathbf{r})=\mathrm{\nabla }\times \mathbf{A}=\frac{{\mu }_0}{4\pi }\left[\frac{3\mathbf{r}(\mathbf{m}\cdot \mathbf{r})}{r^5}-\frac{\mathbf{m}}{r^3}\right].}$

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

${\displaystyle \psi (\mathbf{r})=\frac{\mathbf{m}\cdot \mathbf{r}}{4\pi r^3},}$

and hence the magnetic field strength (or strength of the H-field) is

${\displaystyle \mathbf{H}(\mathbf{r})=-\mathrm{\nabla }\psi =\frac{1}{4\pi }\left[\frac{3\hat{\mathbf{r}}(\mathbf{m}\cdot \hat{\mathbf{r}})-\mathbf{m}}{r^3}\right]=\frac{\mathbf{B}}{{\mu }_0}.}$

The magnetic field strength is symmetric under rotations about the axis of the magnetic moment.In spherical coordinates, with${\displaystyle \hat{\mathbf{z}}=\hat{\mathbf{r}}\cos \theta -\hat{\mathit{\theta }}\sin \theta }$, and with the magnetic moment aligned with the z-axis, then the field strength can more simply be expressed as

${\displaystyle \mathbf{H}(\mathbf{r})=\frac{|\mathbf{m}|}{4\pi r^3}\left(2\cos \theta \hat{\mathbf{r}}+\sin \theta \hat{\mathit{\theta }}\right).}$

The two models for a dipole (current loop and magnetic poles), give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles is 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 (above for mobile users)). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate 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 and area constant, the limiting field is

${\displaystyle \mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\left[\frac{3\hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}{|}^3}+\frac{8\pi }{3}\mathbf{m}\delta (\mathbf{r})\right],}$

whereδ(r) is theDirac delta function in three dimensions. Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.

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

${\displaystyle \mathbf{H}(\mathbf{r})=\frac{1}{4\pi }\left[\frac{3\hat{\mathbf{r}}(\hat{\mathbf{r}}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}{|}^3}-\frac{4\pi }{3}\mathbf{m}\delta (\mathbf{r})\right].}$

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

${\displaystyle \mathbf{M}(\mathbf{r})=\mathbf{m}\delta (\mathbf{r})}$

is themagnetization.

The forceF exerted by one dipole momentm₁ on anotherm₂ separated in space by a vectorr can be calculated using:^[4]

${\displaystyle \mathbf{F}=\mathrm{\nabla }\left({\mathbf{m}}_2\cdot {\mathbf{B}}_1\right),}$

or^[5][6]

${\displaystyle \mathbf{F}(\mathbf{r},{\mathbf{m}}_1,{\mathbf{m}}_2)={\displaystyle \frac{3{\mu }_0}{4\pi r^5}}\left[({\mathbf{m}}_1\cdot \mathbf{r}){\mathbf{m}}_2+({\mathbf{m}}_2\cdot \mathbf{r}){\mathbf{m}}_1+({\mathbf{m}}_1\cdot {\mathbf{m}}_2)\mathbf{r}-{\displaystyle \frac{5({\mathbf{m}}_1\cdot \mathbf{r})({\mathbf{m}}_2\cdot \mathbf{r})}{r^2}}\mathbf{r}\right],}$

wherer is the distance between dipoles. The force acting onm₁ is in the opposite direction.

The torque can be obtained from the formula

${\displaystyle \mathit{\tau }={\mathbf{m}}_2\times {\mathbf{B}}_1.}$

Themagnetic scalar potentialψ produced by a finite source, but external to it, can be represented by amultipole expansion. Each term in the expansion is associated with a characteristicmoment and a potential having a characteristic rate of decrease with distancer from the source. Monopole moments have a1/r rate of decrease, dipole moments have a1/r² rate, quadrupole moments have a1/r³ rate, and so on. The higher the order, the faster the potential drops off. Since the lowest-order term observed in magnetic sources is the dipole term, it dominates at large distances. Therefore, at large distances any magnetic source looks like a dipole of the samemagnetic moment.

• Magnetostatics
• Magnetism
• Electric and magnetic fields in matter

• Articles with short description
• Short description is different from Wikidata