Magnetism and Matter

What this chapter is about

• Chapter 4 showed that moving charges make magnetic fields. Here magnetism is studied in its own right: the bar magnet, Gauss's law for magnetism, and how bulk matter responds to a field.
• Five facts stated in the NCERT introduction, all directly examinable at one mark:
• The earth behaves as a magnet, with its field pointing approximately from geographic south to geographic north.
• A freely suspended bar magnet sets itself north-south; the tip pointing to geographic north is the north pole.
• Like poles repel, unlike poles attract.
• Poles cannot be isolated. Break a bar magnet and you get two complete magnets. Magnetic monopoles do not exist.
• Magnets can be made from iron and its alloys.
• The word "magnet" comes from Magnesia, an island in Greece where magnetic ore was found around 600 BC.

5.2 The bar magnet and its field lines

• Iron filings sprinkled on glass over a short bar magnet arrange themselves in a pattern that mimics the field lines and shows the magnet is a magnetic dipole with a north and a south pole. A current-carrying solenoid gives a very similar pattern.
• Four properties of magnetic field lines — the standard three-mark answer:
• They are continuous closed curves. Outside the magnet they run from N to S; inside the magnet they run from S to N. (Unlike electric field lines of a dipole, which begin on the positive charge and end on the negative charge or escape to infinity.)
• The tangent at any point on a field line gives the direction of the net field $\vec{B}$ at that point.
• The number of lines crossing unit area measures the strength of $\vec{B}$: lines crowd together where the field is strong (near the poles) and spread apart where it is weak (near the middle).
• Two field lines never intersect. If they did, the field would have two directions at the point of intersection, which is impossible.
• Field lines are not "lines of force". The magnetic force on a moving charge is $q\vec{v}\times\vec{B}$, which is perpendicular to $\vec{B}$, so a field line does not show the direction of the force.
• To map the field experimentally, place a small compass needle at various points and note its orientation.

5.2.2 Bar magnet as an equivalent solenoid

• Ampere's hypothesis: all magnetic phenomena can be explained by circulating currents. A bar magnet behaves as a large number of circulating currents, like a solenoid.
• Cutting a bar magnet in half is like cutting a solenoid — two smaller, weaker magnets, each with both poles; the field lines stay continuous.
• Calculating the axial field of a finite solenoid at a far point gives

$$B = \dfrac{\mu_0}{4\pi}\cdot\dfrac{2m}{r^{3}}, \qquad m = n(2l)\,i\,(\pi a^{2}) = NiA$$ which is exactly the far axial field of a bar magnet. Hence the magnetic moment of a bar magnet equals the magnetic moment of the equivalent solenoid producing the same field.

5.2.3 The dipole in a uniform magnetic field

• Torque $\vec{\tau} = \vec{m}\times\vec{B}$, magnitude $\tau = mB\sin\theta$; it is a restoring torque, turning $\vec{m}$ towards $\vec{B}$.
• Net force is zero in a uniform field. So a compass needle turns but does not drift.
• Potential energy $U = -\vec{m}\cdot\vec{B} = -mB\cos\theta$, with the zero chosen at $\theta = 90^\circ$.
• $\theta = 0^\circ$: $U = -mB$, minimum — stable equilibrium, $\tau = 0$.
• $\theta = 90^\circ$: $U = 0$, $\tau = mB$ maximum.
• $\theta = 180^\circ$: $U = +mB$, maximum — unstable equilibrium, $\tau = 0$.
• Work done in rotating from $\theta_1$ to $\theta_2$: $W = -mB(\cos\theta_2 - \cos\theta_1)$.
• An iron nail near a bar magnet feels a force as well as a torque, because the magnet's field there is non-uniform and the nail acquires an induced moment; the attraction arises because the induced pole nearer the magnet is the unlike one.

5.2.4 The electrostatic analogue

• The far field of a magnetic dipole is obtained from the electric dipole result by $\vec{E}\to\vec{B}$, $\vec{p}\to\vec{m}$, $\dfrac{1}{4\pi\varepsilon_0}\to\dfrac{\mu_0}{4\pi}$.

5.3 Gauss's law for magnetism

• Statement: the net magnetic flux through any closed surface is zero, $\phi_B = \sum \vec{B}\cdot\Delta\vec{S} = 0$.
• Reason: magnetic field lines are continuous closed loops, so as many lines enter a closed surface as leave it.
• Significance: isolated magnetic poles (monopoles) do not exist. There are no sources or sinks of $\vec{B}$; the simplest magnetic element is a dipole or a current loop, and all magnetic phenomena are explained by arrangements of dipoles and current loops.
• Contrast with electrostatics: $\sum\vec{E}\cdot\Delta\vec{S} = \dfrac{q}{\varepsilon_0}$ — a non-zero right-hand side, because electric charge exists and can be isolated.
• If monopoles existed the law would read $\oint\vec{B}\cdot d\vec{S} = \mu_0 q_m$, with $q_m$ the enclosed magnetic charge.
• Around a bar magnet's N pole alone, the net flux is still zero — field lines pass through the magnet's interior and close the loops.

5.3 Reading field-line diagrams (the standard "what is wrong" question)

• Lines emanating radially from a point: wrong for magnetism — the net flux through a surface around that point would not be zero. That is the pattern of a long charged wire's electric field.
• Lines crossing: always wrong, for both $\vec{B}$ and $\vec{E}$ — the direction would be ambiguous.
• A closed loop of static magnetic field lines around empty space: wrong. A closed magnetostatic field line must enclose a region through which current passes.
• Lines confined entirely inside a toroid, forming closed loops: right — each loop encloses current.
• Solenoid lines running perfectly straight and staying confined at the ends: wrong; they must curve out and close, or Ampere's law is violated.
• Lines inside and outside a bar magnet forming closed loops: right. Not all lines emanate from the N pole; the net flux around either pole is zero.
• Field lines between two pole pieces being perfectly straight right to the edges: wrong. Some fringing is inevitable.
• Lines all emerging from one plate and ending on another: an electrostatic pattern, impossible for $\vec{B}$.
• A bar magnet exerts no torque on itself due to its own field; nor does a current element feel a force from its own field (though it can feel one from other elements of the same wire).
• A system with zero net charge can still have a magnetic moment — the atomic current loops need not cancel.

5.4 Magnetisation and magnetic intensity

• Magnetisation $\vec{M} = \dfrac{\vec{m}_{\text{net}}}{V}$: net magnetic moment per unit volume, in $\text{A m}^{-1}$.
• Put a core inside a solenoid whose empty-core field is $B_0 = \mu_0 nI$. The field becomes $B = B_0 + B_m$, where $B_m = \mu_0 M$ is the core's contribution.
• Magnetic intensity $\vec{H} = \dfrac{\vec{B}}{\mu_0} - \vec{M}$, so that $\vec{B} = \mu_0(\vec{H}+\vec{M})$. In a solenoid $H = nI$: $H$ carries the external cause (the current), $M$ the material's response.
• For a linear material $\vec{M} = \chi\vec{H}$, giving $B = \mu_0(1+\chi)H = \mu_0\mu_r H = \mu H$.

• $\mu_r$ is the magnetic analogue of the dielectric constant in electrostatics.
• Only one of $\chi$, $\mu_r$, $\mu$ is independent — given one, the other two follow.

5.5 Classifying materials

5.5.1 Diamagnetism

• Diamagnetic substances tend to move from the stronger to the weaker part of an external field: a magnet repels them.
• Their atoms have zero resultant magnetic moment in the absence of a field. Orbiting electrons are equivalent current loops whose orbital moments cancel.
• When a field is applied, electrons whose orbital moment is along the field slow down and those opposed speed up (an induced-current effect, in accordance with Lenz's law). The substance develops a net moment opposite to the applied field, hence the repulsion.
• Inside the specimen the field is reduced, typically by about one part in $10^{5}$; the field lines are expelled.
• Diamagnetism is universal — present in every substance — but so weak that paramagnetism or ferromagnetism masks it where those are present.
• Superconductors are perfect diamagnets: a metal cooled to a very low temperature showing both perfect conductivity and perfect diamagnetism, with the field lines completely expelled, $\chi = -1$, $\mu_r = 0$. This is the Meissner effect. Superconducting magnets are used, for example, in magnetically levitated trains.

5.5.2 Paramagnetism

• Paramagnetic substances get weakly magnetised along an external field and tend to move from weak field to strong field: they are weakly attracted.
• Each atom, ion or molecule has a permanent magnetic dipole moment, but ceaseless random thermal motion keeps them disordered, so there is no net magnetisation without a field.
• In a strong enough field, and at low enough temperature, the atomic dipoles align with $\vec{B}_0$. Raising the field or lowering the temperature increases the magnetisation until saturation, when every dipole is aligned.
• Inside the specimen the field is enhanced, again typically by about one part in $10^{5}$; the field lines crowd in.
• $\chi$ and $\mu_r$ depend on the sample temperature as well as on the material: Curie's law, $\chi = \dfrac{C}{T}$.

5.5.3 Ferromagnetism

• Ferromagnetic substances get strongly magnetised and are strongly attracted from weak field towards strong field.
• Their atoms carry permanent dipole moments as in a paramagnet, but interact so as to align spontaneously over a macroscopic volume called a domain. Typical domain size is about $1\ \text{mm}$, containing roughly $10^{11}$ atoms.
• With no applied field the domain magnetisations point randomly and there is no bulk magnetisation. Applying $\vec{B}_0$ turns the domains towards the field and the favourably oriented domains grow, until the sample becomes a single giant domain. Domain motion is observable under a microscope using a suspension of powdered ferromagnetic material.
• Field lines are highly concentrated inside a ferromagnet; $\mu_r$ exceeds $1000$.
• Hard ferromagnets retain their magnetisation when the field is removed — alnico (iron, aluminium, nickel, cobalt, copper) and naturally occurring lodestone. These make permanent magnets and compass needles.
• Soft ferromagnets lose their magnetisation as soon as the field is removed — soft iron is the standard example.
• Above the Curie temperature $T_C$ the domain structure disintegrates and the ferromagnet becomes a paramagnet, following $\chi \propto \dfrac{1}{T-T_C}$. The loss of magnetisation with temperature is gradual.
• Because $B$ saturates while $H$ keeps rising, $\mu = \dfrac{B}{H}$ for a ferromagnet decreases as the magnetising field is increased.
• Beyond the three main classes there also exist ferrimagnetic, anti-ferromagnetic and spin-glass materials.

Diagrams described in words

• Bar magnet field lines: closed loops leaving the N pole, curving round outside to enter the S pole, and continuing inside the magnet from S to N. Lines crowded near both poles, sparse near the middle. The same picture, drawn for a finite current-carrying solenoid, is indistinguishable at large distances.
• Diamagnetic bar in a uniform field: draw the uniform horizontal lines bending away around the bar, so that inside the bar the lines are fewer and more widely spaced than outside — the specimen expels the field.
• Paramagnetic bar in the same field: the lines bend inwards towards the bar and crowd through it, so the density inside is greater than outside.
• Gaussian surface round one pole of a magnet: draw a closed surface enclosing just the N pole; every line leaving it through the outside is matched by a line entering it through the magnet's body, so the net flux is zero.

Numbers worth remembering

• $\dfrac{\mu_0}{4\pi} = 10^{-7}\ \text{T m A}^{-1}$; $1\ \text{G} = 10^{-4}\ \text{T}$.
• Diamagnetic: $\chi \approx -10^{-5}$. Paramagnetic: $\chi \approx +10^{-5}$. Superconductor: $\chi = -1$, $\mu_r = 0$.
• Copper: $\mu_r = 0.999994$, diamagnetic.
• Ferromagnet: $\mu_r > 1000$.
• Axial field is twice the equatorial field at the same distance, and oppositely directed.

Traps that cost marks

• Do not write that field lines start at N and end at S. They are closed loops: N to S outside, S to N inside. This exact wording has been the answer to a one-mark MCQ.
• A uniform field gives torque but never a net force. Only a non-uniform field (a real bar magnet near a nail) gives a force as well.
• With $\vec{m}$ parallel to $\vec{B}$: torque is zero and force is zero. Do not write $MB$ for either.
• In $W = -mB(\cos\theta_2 - \cos\theta_1)$, $\theta_1$ is the initial and $\theta_2$ the final angle. Getting them the wrong way round flips the sign of the work.
• Stable equilibrium is $\theta = 0^\circ$ with $U = -mB$; unstable is $\theta = 180^\circ$ with $U = +mB$. Both have zero torque, so "torque is zero" alone does not identify stability — quote the energy.
• Gauss's law for magnetism gives zero for every closed surface, not just symmetric ones, and not just ones outside the magnet.
• $\chi$ negative means diamagnetic, however large its magnitude — a value of $-0.5$ is still diamagnetic, since $\mu_r = 1+\chi = 0.5 < 1$.
• Curie's law is $\chi \propto 1/T$ with $T$ in kelvin, and applies to paramagnets. For a ferromagnet above $T_C$ use $\chi \propto 1/(T-T_C)$.
• Above the Curie temperature a ferromagnet becomes paramagnetic, not diamagnetic.
• $H$ inside a solenoid is $nI$ and does not depend on the core. Only $B$ and $M$ change when the core changes.
• Increasing $H$ on a ferromagnet decreases its permeability, because $B$ is approaching saturation.
• Do not talk about isolated pole strengths as though monopoles exist; the rationalised NCERT treats the bar magnet as a dipole equivalent to a solenoid.