Magnetism and Matter

No, and this is the single most important distinction in the chapter. Both $\theta=0°$ and $\theta=180°$ give $\sin\theta=0$, hence $\tau=0$ at both. But they behave oppositely under a small nudge:

• θ = 0° (m ∥ B): $U=-mB$, the minimum possible value of $U$. Nudge $\theta$ slightly away and $U$ rises; a restoring torque appears and pushes the dipole back toward 0°. Stable equilibrium.
• θ = 180° (m antiparallel to B): $U=+mB$, the maximum possible value of $U$. Nudge $\theta$ slightly away and $U$ falls; the torque that appears grows rather than restoring, carrying the dipole all the way around to 0°, never back to 180°. Unstable equilibrium.

The rule: zero torque is necessary for an equilibrium but doesn’t say which kind it is. Check whether $U$ is at a minimum (stable) or a maximum (unstable) right around that point. This is exactly why a compass needle always settles pointing to magnetic north, never south, even though “pointing south” is momentarily also a zero-torque position.

Magnetic field lines never start or stop anywhere. Every one of them is a closed loop: out of a magnet’s north end, round through the space outside, into the south end, and back through the inside of the magnet to where it began (for a solenoid, back through the inside of the coil). There are no magnetic monopoles, no isolated north or south “charges” for a line to begin on or end at.

Now follow one of those closed loops as it meets a closed surface. Every time it crosses outward, it has to cross back inward somewhere else to return to its starting point, so its crossings always come in out-in pairs. Flux counts an outward crossing as positive and an inward one as negative, so each pair cancels, every loop contributes exactly zero, and so does the total over all the loops. (A loop that never touches the surface contributes nothing in the first place.)

The most tempting counterexample is a surface drawn around just the north end of a bar magnet. Lines pour out of it, so surely the net flux is outward? It isn’t, because that surface also cuts through the magnet itself, and inside the magnet the field runs from the south end to the north end, carrying exactly as much flux into the surface as the outside lines carry out. This is also why cutting a magnet in half never isolates a pole: each piece turns out to have its own north and south end, just as cutting a solenoid in two gives two shorter solenoids. The pole-strength model of Section 5.2 is therefore a convenient fiction: it gets the field outside the magnet right, but it would wrongly predict an outward flux through a surface around one end, because it leaves out the field inside the magnet.

Compare this with Gauss’s law for electric fields (see Electric Charges and Fields):

Electric field lines can start and stop: they begin on positive charges and end on negative ones. Wrap a surface around a positive charge and lines leave through it with nothing to bring them back, so the net flux is positive, in proportion to the charge enclosed. Magnetic field lines have no such start or end points, so the right-hand side has nothing to count and is always zero. The entire difference between the two laws is whether or not there is a “charge” for field lines to begin and end on.

One way to see that this is the only difference: if a magnetic monopole of strength $q_m$ were ever found, the magnetic law would gain a right-hand side shaped exactly like the electric one, $\oint \mathbf{B}\cdot d\mathbf{A} = \mu_0 q_m$. Physicists have searched for monopoles, and none has ever been found, so the right-hand side stays at zero.

The angle between geographic north and the direction a compass needle actually points (magnetic north), measured in the horizontal plane. It varies from place to place on the Earth’s surface.

The angle the Earth’s total field makes with the horizontal plane at a location. Near the magnetic equator the field is nearly horizontal (dip ≈ 0°); near the magnetic poles it points almost straight down into the Earth (dip ≈ 90°).

The part of the Earth’s total field $B_E$ lying in the horizontal plane, the component a horizontal compass needle actually responds to:

where $Z_E$ is the vertical component. Together, $H_E$ and $Z_E$ (plus declination, which fixes the horizontal direction) completely specify the Earth’s field at a point. (The subscript $E$ keeps this horizontal component from being confused with the magnetic intensity $H$ of Section 5.6, a different quantity that happens to share the letter.)

Every atom contains electrons circulating around its nucleus, and each circulating electron is a tiny current loop with a tiny magnetic dipole moment. In a bulk sample these moments add as vectors. With no applied field they cancel overall (either each atom’s moment is zero to begin with, or the atoms’ moments point every which way). An applied field upsets that balance, partly lining moments up or inducing small new ones, and leaves the sample with a net moment $m_{\text{net}}$. The magnetisation of the sample is that net moment per unit volume:

It is a vector, pointing along the net moment, and its unit is A·m² per m³, i.e. A/m.

Picture a long rod, length $L$ and cross-sectional area $A$, magnetised uniformly along its length with magnetisation $M$. Its total dipole moment is $M$ times its volume, $M(AL)$.

Microscopically, that moment comes from countless atomic current loops, all circulating the same way round the rod’s axis. Wherever two neighbouring loops touch, their currents run in opposite directions and cancel. Only at the rod’s curved outer surface does a loop have no neighbour to cancel against, so the net effect of all those atomic currents is a thin sheet of current circulating round the outside of the rod: exactly what the windings of a solenoid look like. Call this the magnetising current, $K$ amperes for each metre of the rod’s length.

A solenoid of length $L$ carrying $K$ amperes per metre has $KL$ ampere-turns wrapped round area $A$, so by $m=NIA$ (Section 5.1) its moment is $(KL)A$. That has to be the rod’s moment:

So the magnetisation is the surface current per unit length. The field inside a long solenoid is $\mu_0 nI$, that is, $\mu_0$ times the current per unit length (see Moving Charges and Magnetism), so the field the material contributes is:

Now fill a long solenoid ($n$ turns per metre, current $I$) with the magnetised material. The total field inside is the coil’s own field plus the material’s:

The first term in the bracket, $nI$, depends only on the current we send through the windings; it is the same whether the coil is empty or packed with iron. That quantity is the magnetic intensity $H$ (here, $H = nI$). Its general definition is

which rearranges into the key relation of this section:

$H$ has the same unit as $M$, A/m. A useful way to hold the two apart: $H$ is the cause, set by the free currents in the wires that we control, and $M$ is the material’s response to it. In empty space $M=0$ and the relation shrinks to $B=\mu_0H$.

How strongly does a material respond? For most materials $M$ is simply proportional to $H$, and the constant of proportionality is the magnetic susceptibility:

a pure number, since $M$ and $H$ are both in A/m. Substituting $M=\chi H$ into $B=\mu_0(H+M)$:

where $\mu_r$, the relative permeability, is the factor by which filling the coil with the material multiplies the field (the empty coil gives $B_0=\mu_0H$). So the sign and size of $\chi$ tell you directly what a material does to the field, and they sort materials into the three families of Section 5.7:

• Diamagnetic, χ small and negative (around $-10^{-5}$): $M$ points against $H$, so $B$ ends up a hair smaller than $\mu_0H$, by about one part in $10^5$. The extreme case, $\chi=-1$, gives $B=0$: the field is cancelled completely, which is what happens inside a superconductor.
• Paramagnetic, χ small and positive (around $10^{-5}$ to $10^{-3}$): $M$ points along $H$, and $B$ is a hair larger than $\mu_0H$.
• Ferromagnetic, χ large and positive (around $10^2$ to $10^5$): $M$ is hundreds to many thousands of times bigger than $H$, so almost all of $B$ comes from the material, not the coil. For these materials $M$ is not even strictly proportional to $H$: it depends on the sample’s history, as Section 5.8 shows, so a single value of $\chi$ is only a rough guide.

Small, negative $\chi$ (of order $-10^{-5}$). Weakly repelled by an external field; every material shows this effect at some level, but it’s usually masked by stronger effects when present. Examples: bismuth, copper, water.

Small, positive $\chi$ (of order $10^{-3}$ to $10^{-5}$). Weakly attracted into a field, because each atom carries its own small permanent dipole moment, which thermal motion keeps mostly randomised. Examples: aluminium, sodium, oxygen gas.

Susceptibility falls as temperature rises, since more thermal agitation disrupts the partial alignment a field can induce. This is Curie’s law:

Large, positive $\chi$ (of order $10^2$ to $10^5$). Strongly attracted into a field, and can retain magnetisation after the external field is removed (this is what makes permanent magnets possible; Section 5.8 explains how, and why some ferromagnets hold on far better than others). Examples: iron, cobalt, nickel. Above a material-specific Curie temperature, thermal agitation overcomes the internal alignment entirely, and the material becomes merely paramagnetic.

Put an unmagnetised ferromagnetic rod inside a long solenoid. The current sets $H=nI$ (reversing the current reverses $H$), and a probe measures $B$ inside the rod. Now take $H$ through a full cycle:

1. Up from zero. As $H$ increases, $B$ climbs steeply at first, while domain walls slide easily, then levels off. Eventually every domain is aligned and $M$ can grow no further. This is saturation: from here on, $B$ creeps up only by the small $\mu_0\Delta H$ added by the coil itself.
2. Back down to H = 0. $B$ does not come back down the same curve. Domain walls that moved on the way up get snagged on impurities and defects in the crystal, so many domains stay aligned. With the current switched off completely, $B$ is still well above zero. This leftover field is the retentivity (or remanence), $B_r$: the rod has become a magnet.
3. Reverse the current. It takes a field in the opposite direction to break the alignment up again. $B$ only falls to zero once the reverse field reaches $H=-H_c$. The size of that reverse field, $H_c$, is the coercivity.
4. Keep going, then come back. Pushing $H$ further negative saturates the rod the other way round. Bringing $H$ back up through zero to positive values traces the mirror image ($B=-B_r$ at $H=0$, then $B=0$ at $H=+H_c$) and closes the loop at the original saturation point. The first climb from the unmagnetised state is never repeated: from now on the rod goes round the loop on every cycle.

“Hysteresis” comes from the Greek for “lagging behind”, and that is exactly what $B$ does: on every half-cycle, $H$ passes through zero first and $B$ only gets there later. It also means a ferromagnet has no single susceptibility. At the point $H=0$, $B=B_r$, the magnetisation is large while $H$ is zero, so $M/H$ has no sensible value at all.

Two numbers read straight off the loop describe a ferromagnet for practical purposes. The retentivity $B_r$, where the loop crosses the $B$-axis, says how strong a field the material keeps once the magnetising field is gone. The coercivity $H_c$, where the loop crosses the $H$-axis, says how hard that leftover magnetism is to destroy: how big a reverse field it takes to wipe it out.

The loop’s area matters too. Every trip round the loop drags the domain walls back and forth past the snags that hold them, rather like dragging furniture across a rough floor, and the work done against that resistance ends up as heat in the material. The energy lost per unit volume in one cycle equals the area enclosed by the loop. (The units agree: $H$ in A/m times $B$ in T, where 1 T = 1 N/(A·m) from $F=BIL$, gives N/m², which is the same as J/m³, an energy per unit volume.) A fat loop wastes a lot of energy every cycle; a thin one wastes very little.

Ferromagnets come in two broad kinds, told apart by the shape of their loops. Soft ferromagnets such as soft iron have a tall, narrow loop: a steep climb (a small $H$ produces a large $B$, i.e. a high permeability), a very small coercivity, and a small loop area. Hard ferromagnets such as steel and alnico (an alloy of iron with aluminium, nickel, cobalt, and copper) have a wide, fat loop: a coercivity tens to hundreds of times larger, and a large area. (The current NCERT textbook names these two kinds without drawing the loop; the loop is what explains the difference.) Each job calls for one shape or the other:

• Electromagnets → soft iron. An electromagnet (a scrapyard lifting magnet, a relay, an electric bell) must be strong when switched on and must let go when switched off. The steep climb gives a strong field from a modest current. The tiny coercivity means any leftover magnetism is wiped out by the slightest reverse field, even the weak reverse field that the core’s own poles set up inside it, so when the current stops, the core loses almost all its magnetism. A scrapyard magnet with a steel core would keep hold of some of its load even after the current was cut.
• Transformer cores → soft iron. A transformer core (like the core of a motor or generator) is magnetised one way, then the other, 50 times every second on Indian mains supply, so it goes all the way round its loop 50 times a second. Each lap turns energy equal to the loop area (per unit volume) into heat. A thin loop keeps that waste small and stops the core from overheating; with a fat loop, the core would heat up and waste far more energy every second. The high permeability also means a small current produces a large flux.
• Permanent magnets → steel or alnico. A permanent magnet (a compass needle, a loudspeaker magnet) is magnetised once and must then stay magnetised for years. A high retentivity makes the field it keeps strong; a high coercivity makes that field survive stray fields, knocks, and temperature changes, since only a large reverse field can undo it. The large loop area costs nothing here, because the magnet is never taken round the loop again once it has been made.

One trap worth knowing: soft iron’s retentivity, read off its loop, can be comparable to steel’s or even slightly higher. What rules soft iron out for permanent magnets is its tiny coercivity, not its retentivity: whatever magnetism it keeps is far too easy to destroy. When choosing a permanent-magnet material, coercivity is the property to check first.