Magnetic Properties of Materials

Magnetisation and magnetic intensity — the definitions this section rests on

• A circulating electron in an atom has a magnetic moment. In bulk matter these moments add vectorially and may leave a non-zero total.
• Magnetisation, the net magnetic moment per unit volume:

$$\vec{M} = \dfrac{\vec{m}_{\text{net}}}{V}$$

$\vec{M}$ is a vector with dimensions $[\text{L}^{-1}\text{A}]$, measured in $\text{A m}^{-1}$.

• Take a long solenoid, $n$ turns per unit length, current $I$. With no core the interior field is $B_0 = \mu_0 nI$. Fill the interior with a magnetised material and the field becomes

$$B = B_0 + B_m, \qquad B_m = \mu_0 M$$

where $B_m$ is the contribution of the core and $\mu_0$ is the same constant that appears in the Biot-Savart law.

• Magnetic intensity (magnetising field) is defined by

$$\vec{H} = \dfrac{\vec{B}}{\mu_0} - \vec{M} \qquad\Longleftrightarrow\qquad \vec{B} = \mu_0(\vec{H}+\vec{M})$$

$\vec{H}$ has the same dimensions and units as $\vec{M}$, namely $\text{A m}^{-1}$.

• The split is deliberate: $\vec{H}$ carries the external cause (the current in the windings, $H = nI$ inside a solenoid), while $\vec{M}$ carries the material's own response.
• For a linear material the response is proportional to the cause:

$$\vec{M} = \chi\vec{H}$$

$\chi$, the magnetic susceptibility, is a dimensionless measure of how the material responds. It is small and positive for paramagnetic materials, and small and negative for diamagnetic materials — in the latter case $\vec{M}$ and $\vec{H}$ point in opposite directions.

• Substituting back,

$$\vec{B} = \mu_0(1+\chi)\vec{H} = \mu_0\mu_r\vec{H} = \mu\vec{H}$$

$$\mu_r = 1+\chi, \qquad \mu = \mu_0\mu_r = \mu_0(1+\chi)$$

$\mu_r$ is the relative magnetic permeability, the magnetic analogue of the dielectric constant; $\mu$ is the permeability, with the units of $\mu_0$. Only one of $\chi$, $\mu_r$, $\mu$ is independent — given any one, the other two follow.

• Worked pattern (NCERT Example 5.5): a solenoid with $\mu_r = 400$, $n = 1000\ \text{m}^{-1}$, $I = 2\ \text{A}$. Then $H = nI = 2\times10^{3}\ \text{A m}^{-1}$; $B = \mu_r\mu_0 H = 400\times4\pi\times10^{-7}\times2\times10^{3} = 1.0\ \text{T}$; $M = (\mu_r-1)H = 399\times2\times10^{3} \approx 8\times10^{5}\ \text{A m}^{-1}$; and from $B = \mu_0 n(I+I_M)$, the magnetising current is $I_M \approx 794\ \text{A}$.

Classification by susceptibility

A material is diamagnetic if $\chi$ is negative, paramagnetic if $\chi$ is positive and small, and ferromagnetic if $\chi$ is large and positive.

Here $\epsilon$ is a small positive number. Typically $\chi \approx -10^{-5}$ for a diamagnet and $\chi \approx +10^{-5}$ for a paramagnet — a minuscule difference in $\chi$ produces radically different behaviour.

Diamagnetism

• Definition: diamagnetic substances are those which tend to move from the stronger to the weaker part of an external magnetic field. Where a magnet attracts iron, it repels a diamagnetic substance.
• Origin: the resultant magnetic moment of an atom of a diamagnetic substance is zero in the absence of a field — the orbital moments of the electrons cancel.
• On applying a field, electrons whose orbital magnetic moment is along the field slow down, and those opposed speed up. This is an induced-current effect, in accordance with Lenz's law. The substance therefore develops a net magnetic moment opposite to the applied field, and is repelled.
• Field pattern: the lines are repelled or expelled and the field inside the material is reduced — in most cases by about one part in $10^{5}$.
• In a non-uniform field the bar moves from high field towards low field.
• Examples: bismuth, copper, lead, silicon, nitrogen at STP, water, sodium chloride.
• Diamagnetism is present in all substances, but it is so weak that paramagnetic or ferromagnetic effects usually swamp it.
• Superconductors are the most exotic diamagnets: metals cooled to very low temperatures showing both perfect conductivity and perfect diamagnetism. The field lines are completely expelled, $\chi = -1$ and $\mu_r = 0$. This perfect diamagnetism is the Meissner effect. A superconductor repels a magnet and, by Newton's third law, is repelled by it — the basis of magnetically levitated superfast trains.

Paramagnetism

• Definition: paramagnetic substances are those which get weakly magnetised when placed in an external magnetic field, and tend to move from a region of weak field to a region of strong field — they are weakly attracted.
• Origin: each atom, ion or molecule possesses a permanent magnetic dipole moment of its own. Ceaseless random thermal motion keeps these moments disordered, so no net magnetisation is seen without a field.
• In a sufficiently strong field, and at low temperature, the individual atomic dipoles align along $\vec{B}_0$. Increasing the field or lowering the temperature raises the magnetisation until it reaches its saturation value, at which point every dipole is perfectly aligned with the field.
• Field pattern: the lines get concentrated inside the material and the field inside is enhanced — in most cases by about one part in $10^{5}$.
• Examples: aluminium, sodium, calcium, oxygen at STP, copper chloride.
• Both $\chi$ and $\mu_r$ depend not only on the material but also on the sample temperature, through Curie's law:

$$\chi = \dfrac{C}{T} \qquad\Longrightarrow\qquad \dfrac{\chi_1}{\chi_2} = \dfrac{T_2}{T_1}$$

with $T$ the absolute temperature in kelvin and $C$ the Curie constant. Worked pattern the board has set: magnesium with $\chi = 1.2\times10^{-5}$ at $300\ \text{K}$ reaches $\chi = 1.44\times10^{-5}$ at $T_2 = \dfrac{1.2\times10^{-5}}{1.44\times10^{-5}}\times300 = 250\ \text{K}$.

Ferromagnetism

• Definition: ferromagnetic substances are those which get strongly magnetised when placed in an external magnetic field, and have a strong tendency to move from a region of weak field to a region of strong field — they are strongly attracted.
• Origin: the atoms carry permanent dipole moments as in a paramagnet, but they interact so that they spontaneously align in a common direction over a macroscopic volume called a domain. (Explaining the cooperative effect needs quantum mechanics and is beyond the syllabus.) A typical domain is about $1\ \text{mm}$ across and contains around $10^{11}$ atoms.
• Initially each domain has its own magnetisation, oriented randomly from domain to domain, so there is no bulk magnetisation. On applying $\vec{B}_0$ the domains turn towards $\vec{B}_0$ and, at the same time, the domains already oriented along $\vec{B}_0$ grow at the expense of the others, until the sample becomes one giant aligned domain. Domain motion is not speculation — it can be watched under a microscope after sprinkling a liquid suspension of powdered ferromagnetic material on the sample.
• Field pattern: the field lines are highly concentrated inside the material; in a non-uniform field the sample moves strongly towards the region of high field. The relative permeability exceeds $1000$.
• Hard ferromagnets keep their magnetisation after the external field is removed — alnico (an alloy of iron, aluminium, nickel, cobalt and copper) and the naturally occurring lodestone. These are used as permanent magnets, for example as compass needles. Substances that retain their ferromagnetic property at room temperature for a long period are permanent magnets.
• Soft ferromagnets lose their magnetisation as soon as the external field is removed; soft iron is the standard example.
• Ferromagnetic elements: iron, cobalt, nickel, gadolinium.
• Temperature: the ferromagnetic property depends on temperature. At a high enough temperature the domain structure disintegrates and the ferromagnet becomes a paramagnet; the disappearance of magnetisation with temperature is gradual. Above the Curie temperature $T_C$,

$$\chi \propto \dfrac{1}{T-T_C} \qquad (T > T_C)$$

• Permeability of a ferromagnet is not a constant. Since $\mu = \dfrac{B}{H}$ and $B$ tends towards saturation as $H$ is increased, $\mu$ decreases when the magnetising field is increased. The relation between $B$ and $H$ is non-linear.
• Beyond these three classes there also exist ferrimagnetic, anti-ferromagnetic and spin-glass materials.

Comparison table for the three- and five-mark answers

Diagram: field lines through a specimen

Draw a uniform horizontal field as equally spaced parallel lines with arrows, and place a rectangular bar in the middle.

• Diamagnetic bar: the lines bend away from the bar and go round it, so inside the bar they are fewer and more widely spaced than outside. The specimen expels the field and is weakly repelled.
• Paramagnetic bar: the lines bend towards the bar and crowd through it, so inside the bar they are closer together than outside. The specimen draws the field in and is weakly attracted.
• Ferromagnetic bar: the same as paramagnetic but far more pronounced — almost all the lines in the neighbourhood are funnelled through the specimen.
• The explanation to write underneath: the difference is one of relative permeability. For a diamagnetic substance $\mu_r < 1$ ($\chi_m$ negative), so the lines of force do not prefer to pass through it; for a paramagnetic substance $\mu_r > 1$ ($\chi_m$ positive), so the lines of force prefer to pass through it.

Identifying a material from a single number

• Copper, water, silicon, sodium chloride and bismuth are diamagnetic; aluminium, sodium and copper chloride are paramagnetic; nickel and alnico are ferromagnetic. These exact substances appear in the board's one-mark options.
• A diamagnetic substance is the one whose permeability is less than that of free space, and the only one whose relative permeability lies between $0$ and $1$.

Points that decide one-mark questions

• Magnetic field lines are repelled by a diamagnetic substance.
• Diamagnetic atoms have no resultant magnetic moment in the absence of an external field; paramagnetic atoms do.
• A diamagnetic specimen moves from stronger field to weaker field; a paramagnetic specimen moves from weaker field to stronger field.
• Above the Curie temperature a ferromagnetic material becomes paramagnetic — not diamagnetic.
• Increasing the magnetising field on a ferromagnet decreases its permeability, because $B$ saturates while $H$ keeps rising.
• Diamagnetic susceptibility is essentially independent of temperature; paramagnetic susceptibility follows $\chi \propto 1/T$.