Magnetic properties of materials

Unit 16 treated magnetic fields in empty space. Real coils, however, are wound on iron, and the iron changes everything: the same current produces a field a thousand times stronger, and the material remembers what you did to it. This unit introduces the two field vectors \vec{B} and \vec{H}, the numbers that link them (permeability and susceptibility), the loop that describes magnetic memory (hysteresis), and the three-way classification of all matter as diamagnetic, paramagnetic or ferromagnetic.

17.1 Magnetic field lines and magnetic flux

A magnetic field line is a curve whose tangent at every point gives the direction of the magnetic flux density \vec{B} at that point. Because there are no magnetic monopoles, every field line closes on itself: outside a magnet it runs N to S, inside the magnet it continues S to N.

Rearranged, B = \Phi/A for a surface held perpendicular to the field, which is why B is called the magnetic flux density — flux per unit area. Since every line entering a closed surface must leave it,

This is Gauss's law for magnetism, and it holds inside matter as well as in vacuum. Its practical consequence is that flux is continuous: if a magnetic circuit narrows, the flux stays the same and B rises.

Magnetic materials are studied with a toroidal ring (a Rowland ring) because in a toroid the flux is trapped entirely inside the core, with no leakage and no end effects. The whole of the winding's magnetising action is then felt by the specimen.

17.2 Flux density in magnetic material; Relative permeability; Susceptibility

When a specimen is placed inside a current-carrying solenoid, two separate causes produce the field inside it:

1. the free current in the winding, described by the magnetising field \vec{H}, and
2. the atomic currents inside the specimen itself, described by the intensity of magnetisation \vec{M}.

The resultant flux density inside the specimen is the sum of the two contributions:

Quantity	Symbol	Defining relation	SI unit
Magnetic flux	\Phi	BA\cos\theta	weber (Wb)
Flux density	B	\Phi/A	tesla (T)
Magnetising field	H	NI/l	A m⁻¹
Magnetisation	M	m/V	A m⁻¹
Susceptibility	\chi_m	M/H	none
Permeability	\mu	B/H	T m A⁻¹ (= H m⁻¹)
Relative permeability	\mu_r	\mu/\mu_0	none

Here \mu_0 = 4\pi\times10^{-7}\ \text{T m A}^{-1} is the permeability of free space; for vacuum \chi_m = 0 and \mu_r = 1 exactly.

17.3 Hysteresis

Take an unmagnetised ferromagnetic ring and raise H from zero. B climbs along the initial magnetisation curve Oa and then flattens: all the domains are aligned and the specimen has reached saturation. Now reduce H back to zero. B does not retrace the curve — it falls along ab and stops at a non-zero value B_r. The specimen has become a magnet on its own.

Continuing to reverse H drives the specimen to saturation the other way, and bringing H forward again closes the curve. The closed figure abcdefa is the hysteresis loop.

Energy lost per cycle

Soft and hard magnetic materials

Property	Soft magnetic material	Hard magnetic material
Loop shape	narrow, tall, small area	broad, large area
Retentivity B_r	high	high
Coercivity H_c	very low	very high
Susceptibility, \mu_r	very large	moderate
Energy loss per cycle	small	large
Magnetisation	easily magnetised and demagnetised	hard to demagnetise
Examples	soft iron, silicon steel, mu-metal, ferrites	carbon steel, cobalt steel, alnico, ferrite magnets
Used for	transformer and motor cores, electromagnets, relays	permanent magnets, loudspeakers, compass needles, magnetic latches

17.4 Dia-, para- and ferro-magnetic materials

Every electron in an atom is both orbiting and spinning, and each motion is a tiny current loop with a magnetic moment. How those moments add up decides the class of the material.

Diamagnetic materials. The atomic moments cancel exactly, so the atom has no permanent magnetic moment. An applied field changes the electron orbits slightly (Lenz's law at atomic scale) and induces a moment opposing the field. Hence M is antiparallel to H, \chi_m is small and negative (about -10^{-5}), and \mu_r is slightly less than 1. Field lines are pushed out of the specimen, and a rod suspended in a non-uniform field moves from strong to weak field and sets perpendicular to it. Examples: bismuth, copper, water, gold, mercury, nitrogen, sodium chloride. Superconductors are perfect diamagnets with \chi_m = -1.

Paramagnetic materials. The atoms do have permanent moments, but thermal agitation keeps them randomly oriented. An applied field aligns them slightly, so M is parallel to H with \chi_m small and positive (about 10^{-5} to 10^{-3}) and \mu_r slightly greater than 1. A rod moves from weak to strong field and sets parallel to it. Alignment is destroyed by heat, so susceptibility obeys Curie's law:

where C is the Curie constant. Examples: aluminium, platinum, manganese, chromium, oxygen, copper sulphate solution.

Ferromagnetic materials. Neighbouring atomic moments lock parallel to each other over regions called domains, each containing about 10^{17}–10^{21} atoms. In an unmagnetised sample the domains point in random directions and cancel. A field makes favourably oriented domains grow at the expense of the others and finally rotates them all into line — which is why M saturates. \chi_m is large and positive (10² to 10⁵), \mu_r runs into thousands, field lines crowd strongly into the specimen, and the material shows hysteresis. Examples: iron, cobalt, nickel, gadolinium and alloys such as alnico.

Above a critical temperature, the Curie temperature T_C, thermal energy destroys the domain alignment and a ferromagnet becomes an ordinary paramagnet obeying the Curie–Weiss law:

For iron T_C = 1043\ \text{K} (770\ ^{\circ}\text{C}), for cobalt 1394\ \text{K} and for nickel 631\ \text{K}.

Property	Diamagnetic	Paramagnetic	Ferromagnetic
Atomic moment	zero	permanent, random	permanent, aligned in domains
\chi_m	small, negative (\approx -10^{-5})	small, positive (10^{-5}–10^{-3})	large, positive (10^{2}–10^{5})
\mu_r	slightly < 1	slightly > 1	\gg 1 (10²–10⁵)
M vs H	linear, negative slope	linear, small positive slope	non-linear, saturates
Field lines	expelled	slightly concentrated	strongly concentrated
In non-uniform field	moves to weaker field	moves to stronger field	moves strongly to stronger field
Effect of temperature	almost none	\chi_m = C/T	ferromagnetic below T_C, paramagnetic above
Hysteresis	no	no	yes
Examples	Bi, Cu, Au, H₂O, N₂	Al, Pt, Mn, O₂, CuSO₄	Fe, Co, Ni, Gd, alnico

Chapter summary

• Magnetic flux \Phi = BA\cos\theta (weber); B = \Phi/A is the flux density (tesla); \oint\vec{B}\cdot d\vec{A} = 0 since field lines are closed loops.
• The magnetising field is H = NI/l (A m⁻¹); magnetisation is M = m/V (A m⁻¹); inside matter B = \mu_0(H+M).
• \chi_m = M/H, \mu = B/H, \mu_r = \mu/\mu_0 and \mu_r = 1 + \chi_m. Both \chi_m and \mu_r are dimensionless.
• Hysteresis is the lag of B behind H. Retentivity B_r is the flux density left when H = 0; coercivity H_c is the reverse field that makes B = 0.
• Energy dissipated per unit volume per cycle equals the loop area, \oint H\,dB (J m⁻³); the hysteresis power loss is (area) \times V \times f.
• Soft materials (small H_c, narrow loop) are for transformer cores and electromagnets; hard materials (large H_c, broad loop) are for permanent magnets.
• Diamagnetics: \chi_m small negative, \mu_r < 1, repelled. Paramagnetics: \chi_m small positive, \mu_r > 1, \chi_m = C/T. Ferromagnetics: domains, \chi_m \gg 1, saturation and hysteresis, and \chi_m = C/(T - T_C) above the Curie temperature.

Practice questions

Group A — Multiple choice (1 mark each)

1. The SI unit of magnetic susceptibility is [1]ateslabA m⁻¹cweberdit has no unit
2. For a paramagnetic substance [1]a\chi_m < 0,\ \mu_r < 1b\chi_m > 0,\ \mu_r > 1c\chi_m \gg 1d\mu_r = 1
3. The area of a B–H hysteresis loop represents [1]aretentivitybcoercivitycenergy lost per unit volume per cycledpermeability
4. Soft iron rather than steel is used for a transformer core because it has [1]ahigh coercivityblow coercivity and small loop areaclow permeabilitydhigh density
5. Above its Curie temperature, iron becomes [1]adiamagneticbparamagneticcnon-magneticdmore strongly ferromagnetic

Answers to Group A

1. (d) — \chi_m = M/H, and M and H share the unit A m⁻¹, so the ratio is a pure number.

2. (b) — the moments align weakly with the field, so M is positive and \mu_r = 1+\chi_m > 1.

3. (c) — the loop area is \oint H\,dB, whose unit is J m⁻³.

4. (b) — the core is remagnetised 100 times a second, so the loop area (energy wasted) must be small.

5. (b) — thermal agitation destroys domain alignment, leaving randomly oriented permanent moments.

Group B — Short answer (5 marks each)

1. Define magnetic flux and magnetic flux density, and state their SI units. Distinguish clearly between \vec{B} and \vec{H}. [5]
2. Define magnetic susceptibility and relative permeability, and derive the relation \mu_r = 1 + \chi_m. [5]
3. A solenoid with 400 turns per metre carries a current of 3.0\ \text{A} and is filled with an iron core of relative permeability 800. Calculate H, B, M and \chi_m. [5]
4. Explain, with reference to the hysteresis loop, why soft iron is used in electromagnets while steel is used for permanent magnets. [5]
5. State Curie's law. A paramagnetic sample has \chi_m = 3.6\times10^{-4} at 200\ \text{K}. Find its susceptibility at 300\ \text{K}. [5]

Answers to Group B

3. H = nI = 400\times3.0 = 1.2\times10^{3}\ \text{A m}^{-1}. B = \mu_0\mu_rH = (4\pi\times10^{-7})(800)(1200) = 1.21\ \text{T}. \chi_m = \mu_r - 1 = 799. M = \chi_mH = 799\times1200 = 9.59\times10^{5}\ \text{A m}^{-1}.

4. Outline: an electromagnet must switch on and off, so it needs a small coercivity (demagnetises the instant the current stops) and a small loop area (little heat wasted per cycle) — soft iron. A permanent magnet must keep its magnetism against stray fields and knocks, so it needs a large coercivity and large retentivity — steel. Sketch the narrow and broad loops side by side.

5. Curie's law: for a paramagnetic material \chi_m = C/T, i.e. the susceptibility is inversely proportional to the absolute temperature. \chi_2 = \chi_1T_1/T_2 = 3.6\times10^{-4}\times200/300 = 2.4\times10^{-4}.

Group C — Long answer (8 marks each)

1. (a) What is hysteresis? Draw a complete hysteresis loop for a ferromagnetic material and define retentivity and coercivity on it. [4](b) Show that the energy dissipated per unit volume per cycle equals the area of the loop. A core of volume 0.010\ \text{m}^{3} whose loop area is 300\ \text{J m}^{-3} is used at 50\ \text{Hz}; find the power lost as heat. [4]
2. Compare diamagnetic, paramagnetic and ferromagnetic materials with respect to susceptibility, relative permeability, behaviour of field lines, behaviour in a non-uniform field and effect of temperature, giving two examples of each. Hence explain what happens to a ferromagnetic material as it is heated through its Curie temperature. [8]

Answers to Group C

1. (a) Hysteresis is the lagging of B behind H over a magnetising cycle; B_r (retentivity) is the intercept on the B-axis, H_c (coercivity) the intercept on the H-axis. See the labelled loop in §17.3.

(b) For a ring of volume V = Al and N turns, the source does work dW = eI\,dt = NI\,d\Phi = NI A\,dB = (Al)H\,dB, so W/V = \oint H\,dB = loop area. Numerically, energy per cycle = 300\times0.010 = 3.0\ \text{J}, and

P = Wf = 3.0\times50 = 150\ \text{W}

2. Outline: reproduce the comparison table of §17.4 with two examples each (Bi and Cu; Al and Pt; Fe and Ni). For the Curie point: below T_C the exchange interaction holds neighbouring atomic moments parallel inside domains, giving huge \chi_m, saturation and hysteresis. As T rises, thermal energy disrupts this order; at T = T_C the domain structure collapses completely. Above T_C the atoms keep their individual moments but point randomly, so the material is paramagnetic with \chi_m = C/(T-T_C), a value smaller by several orders of magnitude. For iron T_C = 1043\ \text{K}; a red-hot iron nail is not attracted by a magnet.