Every Magnet Is a Dipole in Disguise

Magnetism is older than recorded science: the word itself traces back to Magnesia, a region of ancient Greece where naturally magnetic ore was found as early as 600 BC, and people have relied on a freely suspended magnet's tendency to point north-south, the working principle of a compass, for many centuries before anyone understood WHY it worked. The previous chapter showed that moving charges, currents, produce magnetic fields; this chapter turns that discovery around, and studies magnetism as a subject of its own, starting with the ordinary bar magnet. A handful of iron filings sprinkled over a bar magnet arranges itself into a pattern strikingly similar to the field lines of an electric dipole, curving from one end of the magnet to the other, immediately suggesting the magnet has two distinct poles, named north and south exactly as their compass-pointing tendency suggests. Several basic, well-known facts about bar magnets hold true: poles of the same kind (north-north or south-south) repel, opposite kinds attract, and no isolated single pole has ever been produced -- break a bar magnet in half and you get two complete, weaker magnets, each with its own north and south pole, not one piece with just a north pole and another with just a south. Tracing these iron-filing patterns into smooth curves gives the magnetic field lines, which obey rules with real substance behind them: they always form continuous, closed loops (unlike electric field lines, which begin and end on charges); the tangent to a field line at any point gives the direction of B there; denser lines mean a stronger field; and, since a magnetic field has one single, definite direction at each point, field lines can never cross.

The resemblance between a bar magnet's field-line pattern and a current-carrying solenoid's is not superficial coincidence, it is deep enough that a bar magnet can genuinely be modelled as a large number of tiny circulating currents, in line with Ampere's own hypothesis that ALL magnetic phenomena ultimately trace back to circulating charge. Cutting a bar magnet in half is, on this view, exactly like cutting a solenoid in half: both operations yield two smaller, weaker versions of the same thing, each still a complete magnetic dipole with its own two poles, never an isolated single pole. The analogy can be made mathematically precise by calculating a finite solenoid's field along its own axis and comparing it with a bar magnet's field measured the same way; at a large distance r from either object's centre (with r much bigger than the object's own size), both give exactly the same result, B = m0(2m)/4pr³, where m is the magnetic moment. Since this expression matches for both the solenoid and the bar magnet, a bar magnet's magnetic moment can be defined simply as whatever moment an equivalent solenoid would need to produce that same far-away field, giving a genuinely useful, physically meaningful way to assign a numerical magnetic moment to an ordinary bar magnet, an object with no obvious current to plug into m=NIA directly.

Place a small compass needle, itself a tiny bar magnet of known magnetic moment m, inside a uniform external magnetic field B, and it experiences a torque, t = m x B, of magnitude mB sinq, where theta is the angle between the needle's own moment and the field, exactly the same relation already found for a current loop, since a bar magnet IS, in effect, a current loop in disguise. This torque is a RESTORING one: it always acts to rotate the needle back toward alignment with the field, which is of course exactly how a compass works. Following the same logic already used for electric dipole potential energy, the magnetic potential energy stored in this configuration is found by integrating the torque over angle, Um = -mB cosq = -m.B, with the zero of potential energy fixed, by convention, at theta=90 degrees, where the needle sits perpendicular to the field. This expression pins down the needle's two extreme orientations precisely: at theta=0, aligned with the field, Um reaches its minimum, most negative value, -mB, the most stable configuration possible; at theta=180 degrees, pointing directly against the field, Um reaches its maximum, +mB, the least stable, an equilibrium that the slightest nudge sends tumbling away from.

Every formula just derived for a bar magnet's torque, potential energy and far field turns out to be an almost exact copy of the corresponding electric dipole formula from earlier in this same textbook, with a small, entirely mechanical set of substitutions: wherever electrostatics has 1/4pe0, magnetism has m0/4p; wherever electrostatics has electric dipole moment p, magnetism has magnetic moment m; and wherever electrostatics has electric field E, magnetism has magnetic field B. Applying this substitution to the electric dipole's known equatorial and axial field expressions gives, immediately, the bar magnet's own far fields: on the equatorial line (the perpendicular bisector of the magnet), BE = -m0m/4pr³; on the axis, BA = m0m/2pr³, exactly twice the equatorial value at the same distance, precisely mirroring the electric dipole's own 2:1 axial-to-equatorial ratio. The torque and potential energy formulas carry over with the same substitution: t = m x B mirrors t = p x E exactly, and Um = -m.B mirrors Um = -p.E exactly. This is not merely a convenient shortcut for remembering formulas, it reflects something genuinely true about the underlying physics: a bar magnet, at large distances, behaves exactly like a mathematical dipole, and every dipole, whether built from electric charges or circulating current, obeys the very same mathematical structure.

Gauss's law for electrostatics relates the electric flux through any closed surface to the charge enclosed within it, positive net flux whenever positive charge is trapped inside. Magnetism has an exact analogue of this law, but with a strikingly different, much simpler right-hand side. Since magnetic field lines always form closed loops, with no beginning and no end, any closed surface drawn anywhere in space, wherever it sits and whatever shape it takes, must have exactly as many field lines entering it as leaving it; the net magnetic flux through it, φB = the sum of B.ΔS over every small area element of the surface, always comes out to precisely zero. This is Gauss's law for magnetism: the net magnetic flux through any closed surface is always zero. The difference between this and its electrostatic cousin is not a minor mathematical detail, it is the direct, formal statement that isolated magnetic poles, monopoles, simply do not exist; there is no magnetic equivalent of a lone electric charge that could sit inside a surface and produce a nonzero net flux. Every magnetic source, without exception, is fundamentally a dipole or a current loop, never a magnetic monopole standing alone, and this law is exactly what would need to be rewritten, replacing zero with m0 times an enclosed magnetic charge, if monopoles were ever actually found.

Real materials placed inside a magnetic field, say, the interior of a current-carrying solenoid, contribute a magnetic response of their own, on top of whatever field the current alone would produce, and untangling these two contributions needs some careful new vocabulary. Every atom's circulating electrons carry a tiny magnetic moment, and in bulk matter, these countless individual moments add up (vectorially) to give the material a net magnetic moment; magnetisation M is defined as this net moment per unit volume, mnet/V. A solenoid with n turns per unit length carrying current I alone produces a field B0 = m0nI; fill its interior with a magnetised material, and the total field becomes B = B0 + Bm, where the material's own extra contribution works out to Bm = m0M. It is convenient to define a further quantity, magnetic intensity H = B/m0 - M, representing the part of the field attributable to external factors alone (the current, in a solenoid), cleanly separated from M, the part contributed by the material itself; rearranged, this gives B = m0(H+M), the total field as the sum of an externally-supplied part and a material-supplied part. For many materials, the material's own response is directly proportional to what is asked of it: M = cH, where c, the magnetic susceptibility, is a dimensionless number describing how strongly (and in which direction) a material responds to a field. Combining these relations gives B = m0(1+c)H = m0mrH = mH, where mr = 1+c is the relative magnetic permeability (magnetism's direct counterpart to the dielectric constant in electrostatics) and m = m0mr is the material's own magnetic permeability; knowing any one of c, mr or m immediately gives the other two.

Diamagnetic substances share one defining tendency: they move from a stronger region of an external field toward a weaker one, meaning an ordinary magnet very slightly REPELS them, exactly the opposite of how a magnet attracts iron. Inside a diamagnetic material, the applied field is genuinely reduced, though usually only very slightly, typically by about one part in one hundred thousand. The mechanism traces back to individual atoms whose own orbiting electrons already sum to exactly zero net magnetic moment; apply an external field, and by the same induction principle explored in the next chapter, electrons orbiting in one sense speed up while electrons orbiting the other way slow down, producing a small net induced moment that opposes, rather than aligns with, the applied field, exactly why the substance is gently pushed away rather than pulled in. Diamagnetism is genuinely universal, present in every substance without exception, including bismuth, copper, lead, water and ordinary salt, but it is so weak that it is almost always masked entirely whenever a stronger paramagnetic or ferromagnetic effect is also present in the same material. The most extreme diamagnets known are superconductors, metals cooled to extremely low temperatures, which expel an applied field completely, achieving c = -1 and mr = 0 exactly; this perfect diamagnetism, called the Meissner effect, is put to genuinely practical use in magnetically levitated trains, where a superconductor's total repulsion of a magnetic field lifts the train clear of its track.

Paramagnetic substances behave oppositely to diamagnetic ones: placed in an external field, they become weakly magnetised in the SAME direction as the field, and so are gently drawn from weaker regions of a field toward stronger ones, the reason a paramagnetic sample feels a faint pull toward a magnet rather than a push away. The mechanism here starts from individual atoms (or ions, or molecules) that already possess a permanent magnetic moment of their own, entirely independent of any external field, unlike the zero-net-moment atoms of a diamagnetic substance. With no field applied, ceaseless random thermal motion keeps these countless individual moments pointing every which way, so the material shows no net magnetisation at all on average; switch on a sufficiently strong external field, and especially at low temperature, where thermal chaos is weaker, the individual moments increasingly align with the field, producing a real, if still typically modest, net magnetisation, again usually only about one part in one hundred thousand under ordinary conditions. This is why paramagnetism, unlike diamagnetism, genuinely depends on temperature: raise the temperature, and thermal agitation increasingly scrambles the alignment faster than the field can restore it, weakening the effect; lower the temperature or strengthen the field, and magnetisation climbs, eventually approaching a saturation value where essentially every atomic moment has fallen into line. Aluminium, sodium, calcium, oxygen and copper chloride are common paramagnetic materials.

Ferromagnetic substances take paramagnetism's basic starting point, individual atoms with their own permanent magnetic moment, and add one further, genuinely dramatic ingredient: neighbouring atomic moments interact strongly enough to spontaneously align with each other, entirely on their own, across whole macroscopic regions called domains, each domain typically about a millimetre across and containing on the order of 10¹¹ atoms, all pointing the same way, with no external field needed to make it happen. (The precise mechanism behind this cooperative alignment needs quantum mechanics to explain properly, beyond the scope of this chapter.) With no external field applied, different domains within a sample point in different, essentially random directions, so the material as a whole shows little or no NET magnetisation despite every individual domain already being strongly magnetised internally. Apply an external field, and the domains already pointing roughly along it grow at the expense of their neighbours, while domains throughout the sample gradually rotate to align, until, at a strong enough field, the entire sample effectively becomes one single giant domain, producing a very large, strongly concentrated field, thousands of times stronger a response than a diamagnetic or paramagnetic material could ever manage, with relative permeability well over a thousand. What happens once that external field is switched back off separates ferromagnetic materials into two practically important categories: 'hard' ferromagnets, such as the alloy alnico or naturally occurring lodestone, retain much of their magnetisation even with no field present, and are exactly what permanent magnets and compass needles are made from; 'soft' ferromagnets, such as ordinary soft iron, lose almost all their magnetisation the instant the external field is removed, useful whenever a magnet needs to be switched on and off at will, such as inside an electromagnet. Ferromagnetism itself is not permanent even for hard magnets under all conditions: heat any ferromagnet enough, and its cooperative domain structure gradually breaks down, until at a high enough temperature it behaves as an ordinary paramagnet instead.