Physics · Electricity and magnetism · Grade 10

Magnetic properties of matter

Why does iron stick to a magnet, bismuth shrink away from it, and hot nickel let go?

The situation

An iron nail sticks to a magnet and an aluminium spoon hardly feels it — but a piece of bismuth hung between the poles of a strong electromagnet recoils from the field, and so does a drop of water. A transformer core is re-magnetised 50 times every second and gets hot, while a permanent magnet stays magnetic for years. If you heat a nickel washer in a flame, it falls away from the magnet; when it cools, it is attracted again. Why do substances respond to a magnetic field so differently?

Topic

Magnetic properties of matter. Magnetising a nail, hysteresis and the weakening of a heated magnet

Key ideas

Diamagnetic, paramagnetic, ferromagnetic
Hysteresis loop
Curie point

Formulas

\vec{B} = μ_{0} (\vec{H} + \vec{M}) = μ_{0} μ \vec{H}, \vec{M} = χ \vec{H}, μ = 1 + χ

Magnetisation vector $\vec{M}$ , magnetic field strength $\vec{H}$ and magnetic permeability $μ$ .

F_{z} = \frac{χ V}{μ_{0}} B \frac{d B}{d z}

Force on a weakly magnetic material in a non-uniform field: $χ < 0$ — repelled from the field, $χ > 0$ — attracted.

χ_{p a r a} = \frac{C}{T}, χ = \frac{C}{T - T_{C}} (T > T_{C})

Curie and Curie–Weiss laws: thermal motion hinders the ordering of the moments.

w_{h y s t} = \oint H d B

The energy converted into heat per unit volume in one magnetisation cycle is the area of the hysteresis loop.

ω_{L} = \frac{e B}{2 m_{e}}

Larmor frequency: the field makes the electron orbitals precess — the cause of diamagnetism.

Theory

Three kinds of response: diamagnets, paramagnets and ferromagnets

If a substance is placed in a field, it becomes magnetised: the magnetic moment per unit volume is $\vec{M} = χ \vec{H}$ , and the resulting field is $\vec{B} = μ_{0} (\vec{H} + \vec{M}) = μ_{0} μ \vec{H}$ . In diamagnets $χ < 0$ and is very small (bismuth $- 1.7 \cdot 10^{- 4}$ , water $- 9 \cdot 10^{- 6}$ , copper $- 1 \cdot 10^{- 5}$ ): they are repelled from where the field is strong. In paramagnets $χ > 0$ and is small (aluminium $2.2 \cdot 10^{- 5}$ ): they are weakly attracted. In ferromagnets (iron, nickel, cobalt) $μ$ is hundreds to hundreds of thousands, non-linear and dependent on history. The deflection is $\tan θ = χ B B^{'} / (μ_{0} ρ g)$ — because the density of water is small, it deflects more than copper. The energy stored in the field also depends on the substance (the energy of the magnetic field — in the “Self-induction” experiment).

Ampère's molecular currents and the elementary theory

Ampère proposed that every molecule contains a closed molecular current — a tiny magnet. In today's language, these are the orbital motion of electrons and their spin; the magnetic moment is proportional to the angular momentum: ${\vec{p}}_{m} = - \frac{e}{2 m_{e}} \vec{L}$ . Paramagnetism: the permanent moments of atoms are partially aligned along the field, while thermal motion hinders this — $χ = C / T$ (Curie's law); for an ideal paramagnet $M = n p_{m} \tanh (p_{m} B / k T)$ . Diamagnetism is present in all substances: the field makes the electron orbitals precess at the Larmor frequency $ω_{L} = e B / (2 m_{e})$ (Larmor's theorem), and the extra current that appears, by Lenz's law, gives a moment opposing the field. In metals the conduction electrons give Pauli paramagnetism (spins) and Landau diamagnetism (orbits, $χ_{L} = - χ_{P} / 3$ ); gases such as oxygen are paramagnetic. See the molecular currents in the “Magnifier”.

Magnetisation vector and magnetic field strength

If the molecular currents are averaged macroscopically, surface and volume currents equivalent to the magnetisation $\vec{M}$ arise ( ${\vec{j}}_{m} = curl \vec{M}$ ) — similar to the polarisation $\vec{P}$ of a dielectric. That is why two vectors are used in matter: $\vec{B}$ (the field of all currents, $\oint \vec{B} \cdot d \vec{S} = 0$ ) and $\vec{H} = \vec{B} / μ_{0} - \vec{M}$ , whose circulation depends only on the free (conduction) currents: $\oint \vec{H} \cdot d \vec{l} = \sum I_{f r e e}$ . For a ring sample, this is exactly where $H = N_{1} I / l$ comes from.

Ferromagnetism: exchange and domains

In a ferromagnet the spins of neighbouring atoms line up in parallel because of the quantum exchange interaction — thousands of times stronger than the interaction of magnetic dipoles. But if the whole sample were magnetised in one direction, the energy of the external field would be large, so it splits spontaneously into magnetised regions — domains (micrometres to millimetres). As the field increases, first the walls of the domains aligned with the field move and these domains grow, then the moments rotate towards the field — saturation. If neighbouring spins line up in opposite directions, the material is antiferromagnetic (MnO); with unequal opposing lattices it is ferrimagnetic (ferrites: strongly magnetised but poor conductors of electric current, so there are few eddy currents in high-frequency cores). At low temperature, small wave-like disturbances of the order — spin waves (magnons) — reduce $M$ .

Hysteresis and obtaining it on an oscilloscope (lab)

In a ferromagnet $B$ lags behind $H$ — hysteresis. The induction left at $H = 0$ is the remanence $B_{r}$ ; the reverse field needed to remove the magnetisation is the coercivity $H_{c}$ . The loop area $\oint H d B$ is the energy converted to heat in each cycle. Soft materials (transformer steel, ferrite) have a narrow loop, $H_{c}$ tens of A/m: they are used for cores. Hard materials have a wide loop, $H_{c}$ tens of thousands of A/m: they are used for permanent magnets. Procedure: the primary coil ( $N_{1}$ ) of the ring sample is connected to alternating current through $R_{1}$ ; $U_{x} = i R_{1} \propto H = N_{1} i / l$ is fed to the X input of the oscilloscope. The EMF of the secondary coil ( $N_{2}$ ) is integrated by the $R_{2} C$ integrator: $U_{y} = N_{2} S B / (R_{2} C) \propto B$ — to the Y input. The loop is drawn on the screen; find $B_{r}$ , $H_{c}$ and $B_{m a x}$ from the scale.

The Curie point and thermodynamics of magnetic materials (lab)

As the temperature rises, thermal motion destroys the order of the spins and the spontaneous magnetisation decreases; at the Curie point $T_{C}$ it vanishes and the material becomes paramagnetic: $χ = C / (T - T_{C})$ . For nickel $T_{C} \approx 358$ °C, for iron ≈ 770 °C, for gadolinium ≈ 20 °C. Procedure: a nickel washer, held to the magnet, is heated in a flame; a thermocouple shows the temperature. The temperature at the moment the washer drops off equals $T_{C}$ ; when it cools in air it is attracted again — repeat the measurement several times and take the average. The link between magnetic order and temperature is also used in cooling: if a paramagnetic salt is magnetised in a strong field and then demagnetised adiabatically, it cools (to millikelvins; with nuclear moments, to microkelvins).

Did you know?

In his 1895 doctoral thesis Pierre Curie studied how the magnetic properties of substances depend on temperature: he showed that the susceptibility of paramagnets is inversely proportional to absolute temperature (Curie's law) and that ferromagnets lose their ferromagnetism above a certain temperature.
In 1997, at the high-field laboratory in Nijmegen in the Netherlands, Andre Geim held a living frog levitating in air in a field of about 16 T: the tiny diamagnetism of water and tissues was enough to overcome its weight.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Electric current in vacuum and gases — Grade 8
Magnetic field — Grade 8
Electromagnet and relay — Grade 8
Shake torch — Grade 10
Self-induction — Grade 10
Plasma — the fourth state — Grade 10
Power transmission line — Grade 11
Microwaves and Wi-Fi — Grade 11
Radiation from a dipole antenna — Grade 11
Oscillating circuit (LC) — Grade 11