### Magnetic poles

Review of basic magnetostatics

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So let¡¯s begin by defining the magnetic field, H, in terms of magnetic poles.This is the order in which things happened historically ¨C the law of interaction between magnetic poles was discovered by Michell in England in 1750, and by Coulomb in France in 1785, a few decades before magnetism was linked to the flow of electric current. These gentlemen found empirically that the force between two magnetic poles is proportional to the product of their pole strengths, p, and inversely proportional to the square of the distance between them,F ¡Øp1p2 r2 . (1.1)

This is analogous to Coulomb¡¯s law for electric charges, with one important difference ¨C scientists believe that single magnetic poles (magnetic monopoles) do not exist. They can, however, be approximated by one end of a very long bar magnet, which is how the experiments were carried out. By convention, the end of a freely suspended * bar magnet* which points towards magnetic north is called the north pole, and the opposite end is called the south pole.1 In cgs units, the constant of proportionality is unity, so

F =p1p2r2 (cgs), (1.2)

where r is in centimeters and F is in dynes. Turning Eq. (1.2) around gives us the definition of pole strength:

A pole of unit strength is one which exerts a force of 1 dyne on another unit pole located at a distance of 1 centimeter.

The unit of pole strength does not have a name in the cgs system. In SI units, the constant of proportionality in Eq. (1.1) is ¦Ì0/4¦Ð, soF =¦Ì04¦Ðp1p2r2 (SI), (1.3)

where ¦Ì0 is called the permeability of free space, and has the value 4¦Ð ¡Á 10−7 weber/(ampere meter) (Wb/(Am)). In SI, the pole strength is measured in ampere meters (A m), the unit of force is of course the newton (N), and 1 newton = 105dyne (dyn).

To understand what causes the force, we can think of the first pole generating a magnetic field, H, which in turn exerts a force on the second pole. SoF =p1r2p2 = Hp2, (1.4)giving, by definition,H =p1r2 . (1.5)So:A field of unit strength is one which exerts a force of 1 dyne on a unit pole.By convention, the north pole is the source of the magnetic field, and the southpole is the sink, so we can sketch the magnetic field lines around a bar magnet asshown in Fig. 1.1.

The units of magnetic field are oersteds (Oe) in cgs units, so a field of unitstrength has an intensity of 1 oersted. In the SI system, the analogous equation forthe force one pole exerts on another is

yielding the expression.

The earth¡¯s magnetic field has an intensity of around one-tenth of an oersted,and the field at the end of a typical kindergarten toy bar magnet is around5000 Oe.

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