Basic Properties of Magnets

Before deriving any formulas, the book fixes five pieces of vocabulary that recur throughout the chapter: what the magnetic dipole moment of a bar magnet means, how the magnetic field itself is defined, how magnets are classified, what magnetic flux measures, and the distinction between a uniform and a non-uniform f …

For a bar magnet with poles of pole strength $q_m$ separated by a magnetic length $2l$ (measured between the two pole points, centred on the geometric centre $O$), the magnetic dipole moment is defined as

where $\vec d$ is the vector drawn from the south pole to the north pole, of magnitude $d = 2l$. In magnitude, $p_m = q_m(2l)$. It is a vector quantity directed from the south pole to the north pole, with SI unit A m$^2$. Pole strength $q_m$ itself is a scalar with dimension $[M^0LT^0A]$ and SI unit N T$^{-1}$ (equivalently A m); like electric charge, the north pole of a magnet feels a force along $\vec B$ while the south pole feels a force opposite to $\vec B$. Pole strength depends on the material, the cross-sectional area, and the state of magnetisation, and it is halved if the magnet is cut along its length but unchanged if it is cut perpendicular to its length -- because cutting perpe …

The magnetic field is the region around a magnet within which its influence can be felt by another magnet placed in that region. Formally, the field $\vec B$ at a point is defined as the force experienced by a hypothetical unit north pole ($q_m = 1$) kept at that point:

Its SI unit is N A$^{-1}$ m$^{-1}$ (equivalent to tesla, T, once current-based units are introduced later in the chapter). Field lines drawn around a magnet run from north to south outside the magnet and from south to north inside it; they are continuous closed curves that never intersect (an intersection would mean the compass needle poi …

Magnets are either natural (iron, cobalt, nickel -- weak and irregularly shaped) or artificial (made to a desired shape and strength; a rectangular or cylindrical artificial magnet is a bar magnet). Bar magnets obey a short list of properties: (1) a freely suspended bar magnet always settles along the north-south direction; (2) it attracts or repels other magnets/magnetic substances most strongly near its ends (iron filings cling densest at the tips); (3) breaking a magnet into pieces always produces new complete magnets, each with its own pair of poles -- isolated monopoles do not exist; (4) the two poles of a magnet have equal pole strength; (5) the geometrical length (the physical end-to-end length) is always slightly longer than the magnetic length (the separation between the actual pole points), with the ratio …

The magnetic flux $\Phi_B$ through an area is the number of magnetic field lines crossing it normally. For a uniform field $\vec B$ through a flat area $\vec A$,

where $\theta$ is the angle between $\vec B$ and the area's normal $\vec A$. It is maximum ($\Phi_B = BA$) when $\vec B$ is normal to the surface ($\theta=0°$) and zero when $\vec B$ lies in the surface ($\theta=90°$). For a non-uniform field, $\Phi_B = \int \vec B \cdot d\vec A$. Magnetic flux is a scalar; its SI unit is the weber (Wb), with dimensional formula $[ML^2T^{-2}A^{-1}]$, and its CGS unit is the maxwell ($1$ Wb $=10^8$ maxwell). The related quantity magnetic flux density -- the number of field lines crossing unit area held normal to them -- has unit Wb m$^{-2}$ or tesla (T). For any closed surface enclosing a magnetic dipole, the net outward f …

A magnetic field is uniform if it has the same magnitude and the same direction at every point of a given region -- the Earth's magnetic field, for instance, is uniform over the small scale of a school or a physics laboratory. A magnetic field is non-uniform if either its magnitude or its direction (or both) changes from point to point in a region; the field of a bar magnet is the standard example, since both the strength and the direction of $\vec B$ change continuously as you move around the magnet. This distinction matters later in the chapter: a magnetic dipole in a uniform field feels only a torque (no ne …