Magnetic dipole

In electromagnetism, a magnetic dipole is a theoretical description of a sufficiently small magnet such as that of an atom or electron. All magnets can be described as being a magnetic dipole for sufficiently large distances from the magnet. The strength of a magnetic dipole is determined by a single property its magnetic dipole moment. The magnetic dipole model accurately predicts many properties of small magnets such as the magnetic field it produces, how it interacts with other magnetic dipoles, and how an external magnetic field will apply a torque or create a net force on the dipole. Two different models can be used to describe a magnetic dipole. The simplest to understand, but least correct, is to imagine the magnet as 2 equal but opposite poles that act similar to electric charges. The 'ideal' magnetic dipole then is modeled by shrinking the distance between the poles while increasing the magnetic pole strength such that product of the two (the magnetic dipole moment) remains at the given value for that dipole. This can often give correct results in an easy to understand way, but suffers from being incorrect (magnetic poles do not exist as separate entities) and giving incorrect results in certain cases (for example inside of a magnet). The more correct description of a magnetic dipole is that of a closed loop of electric current that encloses a flat area a. The magnetic moment of this dipole then is the product of its area and it current. This amperian loop model has the advantage of being physically correct, at least for the part of the magnetic field of an atom due to the motion of the electrons around the nucleus of atoms. 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.

Models The preferred classical explanation of a magnetic dipole has changed over time. Before the 1930s, textbooks explained the magnetic dipole using hypothetical magnetic point charges. Since then, most have defined it in terms of Ampèrian currents. In magnetic materials, the cause of the magnetic moment are the spin and orbital angular momentum states of the electrons.

Magnetic pole model

The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. This is sometimes known as the Gilbert model. In this model, a small magnet is modeled by a pair of fictitious magnetic monopoles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: the strength p of its poles (magnetic pole strength), and the vector ℓ {\displaystyle \mathrm {\boldsymbol {\ell }} } separating them. The magnetic dipole moment m is related to the fictitious poles as

m = p ℓ . {\displaystyle \mathbf {m} =p\,\mathrm {\boldsymbol {\ell }} \,.}

It points in the direction from the South to the North pole. The analogy with electric dipoles should not be taken too far because magnetic dipoles are associated with angular momentum (see Relation to angular momentum). Nevertheless, magnetic poles are very useful for magnetostatic calculations, particularly in applications to ferromagnets. Practitioners using the magnetic pole approach generally represent the magnetic field by the irrotational field H, in analogy to the electric field E.

Ampèrian loop model

After Hans Christian Ørsted discovered that electric currents produce a magnetic field and André-Marie Ampère discovered that electric currents attract and repel each other similar to magnets, it was natural to hypothesize that all magnetic fields are due to electric current loops. In this model developed by Ampère, the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of current I. The dipole moment of this loop is

m = I S , {\displaystyle \mathbf {m} =I{\boldsymbol {S}},}

where S is the area of the loop. The direction of the magnetic moment is in a direction normal to the area enclosed by the current consistent with the direction of the current using the right hand rule.

Localized current distributions

The magnetic dipole moment can be calculated for a localized (does not extend to infinity) current distribution assuming that we know all of the currents involved. Conventionally, the derivation starts from a multipole expansion of the vector potential. This leads to the definition of the magnetic dipole moment as:

m = 1 2 ∭ V r × j ( r ) d V , {\displaystyle \mathbf {m} ={\tfrac {1}{2}}\iiint _{V}\mathbf {r} \times \mathbf {j} (\mathbf {r} )\,\mathrm {d} V,}

where × is the vector cross product, r is the position vector, and j is the electric current density and the integral is a volume integral. When the current density in the integral is replaced by a loop of current I in a plane enclosing an area S then the volume integral becomes a line integral and the resulting dipole moment becomes

m = I S , {\displaystyle \mathbf {m} =I\mathbf {S} ,}

which is how the magnetic dipole moment for an Amperian loop is derived. Practitioners using the current loop model generally represent the magnetic field by the solenoidal field B, analogous to the electrostatic field D.

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