Magnetic dipole

 ( Magnetostatics ) 

In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant.

It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a true magnetic monopole, the magnetic analogue of an electric charge, has never been observed in nature.

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.

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External magnetic field produced by a magnetic dipole moment In classical 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 the magnetic moment constant. For the current loop, this limit is most easily derived from the vector potential:

$\{\backslash mathbf\{A\}\}(\{\backslash mathbf\{r\}\})=\backslash frac\{\backslash mu\_\{0\}\}\{4\backslash pi\}\backslash frac\backslash times\{\backslash mathbf\{r\}\}\}\{r^\{3\}\},$

where μ₀ is the vacuum permeability constant and is the surface of a sphere of radius . The magnetic flux density (strength of the B-field) is then

$\backslash mathbf\{B\}(\{\backslash mathbf\{r\}\})=\backslash nabla\backslash times\{\backslash mathbf\{A\}\}=\backslash frac\{\backslash mu\_\{0\}\}\{4\backslash pi\}\backslash left\backslash frac\{3\backslash mathbf\{r\}(\backslash mathbf\{m\}\backslash cdot\backslash mathbf\{r\})\}\{r^\{5\}\}-\backslash frac\}\{r^\{3\}\}\backslash right.$

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

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

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

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

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

$\backslash mathbf\{H\}(\{\backslash mathbf\{r\}\})=\backslash frac${4\pi r^3} \left (

2 \cos \theta \, \mathbf{\hat{r}} + \sin \theta \, \boldsymbol{\hat{\theta}} \right ) .

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Internal magnetic field of a dipole 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

$\backslash mathbf\{B\}(\backslash mathbf\{r\})=\backslash frac\{\backslash mu\_0\}\{4\backslash pi\}\backslash left\backslash frac\{3\backslash mathbf\{\backslash hat\{r\}\}(\backslash mathbf\{\backslash hat\{r\}\}\backslash cdot,$

where is the Dirac 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

$\backslash mathbf\{H\}(\backslash mathbf\{r\})\; =\backslash frac\{1\}\{4\backslash pi\}\backslash left\backslash frac\{3\backslash mathbf\{\backslash hat\{r\}\}(\backslash mathbf\{\backslash hat\{r\}\}\backslash cdot.$

These fields are related by , where

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

is the magnetization.

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Forces between two magnetic dipoles The force exerted by one dipole moment on another separated in space by a vector can be calculated using:

D.J. Griffiths (2026). 9788177582932, Pearson Education. ISBN 9788177582932

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

or

$$

\mathbf{F}(\mathbf{r}, \mathbf{m}_1, \mathbf{m}_2) = \dfrac{3 \mu_0}{4 \pi r^5}\left(\mathbf{m}_1\cdot\mathbf{r})\mathbf{m}_2,

where is the distance between dipoles. The force acting on is in the opposite direction.

The torque can be obtained from the formula

$\backslash boldsymbol\{\backslash tau\}=\backslash mathbf\{m\}\_2\; \backslash times\; \backslash mathbf\{B\}\_1.$

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Dipolar fields from finite sources The magnetic scalar potential produced by a finite source, but external to it, can be represented by a multipole expansion. Each term in the expansion is associated with a characteristic Multipole moment and a potential having a characteristic rate of decrease with distance from the source. Monopole moments have a rate of decrease, dipole moments have a rate, quadrupole moments have a 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 same magnetic moment.

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Notes

• Tai L. Chow (2026). 9780763738273, Jones & Bartlett Learning. . ISBN 9780763738273
• (1975). 047143132X, Wiley. . 047143132X
• Edward P. Furlani (2026). 9780122699511, Academic Press. . ISBN 9780122699511

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