Lecture 7b Magnetism

1. Lecture 7bMagnetism Contents • 7.5 Magnetic properties of materials • 7.6 Soft ferromagnetic materials • 7.7 Hard ferromagnetic materials • 7.8 Paramagnetism and diamagnetism

2. 7.5 Magnetic properties of materials • So far we assumed that our conductors are surrounded by vacuum when calculating B fields • But coils nearlyalways have iron cores to: • Increase B • Confine B to desired regions • Just as we adapted Gauss’s law for electric fields in dielectrics (lecture 3b) we must adapt Ampere’s law for magnetic fields in materials. • There are close parallels between the electric and magnetic properties of materials and dipoles play a crucial part in both.

3. 7.6 Soft ferromagnetics • Pure iron and also some iron alloys such as silicon iron (as used in transformers) and more particularly some nickel iron alloys such as permalloy and mumetal are very easily magnetised. • Small applied magnetic fields will line up all the participating electron spin dipole moments inside the material. • Magnetisation (magnetic moment per unit volume) • M =  / V • In ferromagnets M is ‘soft’ in since it nearly all disappears if the applied field is just slightly reversed.

4. N S s N 7.6 Soft ferromagnetics • electron dipoles line up with the external field B. • Individual dipoles have returning fields around them which are in the opposite direction to the applied B • Volume averaged field inside is much less than outside. • cf. E in electrostatics

5. 7.6 Soft ferromagnetics • Volume averaging the dipole vector B fields inside the material gives • Rename the field inside

6. 7.6 Soft ferromagnetics • Our first equation now becomes • B is continuous across the boundaries and we have a new quantity. H = magnetic intensity • For a continuous ring of mu-metal around a current-carrying wire, Ampere’s law 

7. 7.6 Soft ferromagnetics • M and H have the same units (A/m). where χm is the magnetic susceptibility For mu-metal χm is typically 3×104. where dimensionless μr is called the relative permittivity which in mu-metal may be about 3×104. (magnetic equivalent of dielectric constant)

8. 7.6 Materials: the process of magnetisation • Homogeneous material is a simplified model • Usually polycrystalline – a mosaic of tiny perfect crystals • The electrons usually line up spontaneously in each crystal (domain) even without an applied field along one of the preferred crystal axes. Unmagnetised state

9. 7.6 Materials: the process of magnetisation • Within each domain, atomic magnetic moments parallel. Domain magnetisations randomly oriented. • Domains tend to orient themselves parallel to B and boundaries shift. • Domains magnetised parallel to the field grow whilst others shrink • Within each

10. Soft ferromagnetic material Narrow hysteresis loop – less energy dissipation. Good for electromagnets, transformers, motors and generators. ( Eg. soft iron) Hard ferromagnetic material Broad hysteresis loop – need large reverse field to demagnetise. Good for permanent magnets. Eg Steels and alloys (Alnico) 7.7 Materials: soft and hard ferromagnetics • Hysteresis loops. • Magnetising and demagnetising dissipates energy resulting in temperature increase.

11. 7.7 Super-hard ferromagnetics • Some modern materials are so magnetically hard that they cannot be demagnetised except by special methods. The B achieved in a wide-gap magnet design can be an order of magnitude higher than with the older materials. • eg Samarium-Cobalt and neodymium-iron

12. 7.7 Super-hard ferromagnetics • To make this material, a powder of tiny needle-shaped crystals of hexagonal samarium-cobalt is mixed with liquid epoxy to make a paste. • Put into a mould in a strong B field of several tesla. N • The applied B lines up all the crystals so that they have their c axes and N poles pointing towards the top. • When the epoxy sets the block is removed from the mould and B field.

13. 7.7 Super-hard ferromagnetics • Why is this such a powerful permanent magnet? • Needle-shaped crystals find it much more energetically favourable for their magnetisation to be along the length rather than across the width. • The crystal grains (domains) are kept slightly apart by the epoxy so they don’t clump and so are guaranteed to behave as individuals.

14. 7.8 Paramagnetism • For many materials, each atom has a net magnetic moment. • ie Permanent atomic magnets due to electron spin, electon orbital circulation and nuclear magnetic moments • An external B field exerts a torque on each moment • tends to align moments with the field (minimum P.E.) • Current loops from each magnetic moment addto the external field B. • Paramagnetic fields increase the field in the material • m > 0(in the range than 10−3 to 10−6)

15. 7.8 Paramagnetism

16. 7.8 Curie’s Law • Random thermal motion opposes the alignment. • Paramagnetic susceptibility decreases with increasing temperature • M = C ( B / T) • C = Curie’s constant – depends on material • Magnetic dipoles are attracted to the poles of a magnet But….thermal randomization makes attraction for paramagnets very weak. Eg. Aluminium which you can’t pick up with a magnet

17. 7.8 Diamagnetism • Applies to all materials, even those where the total magnetic moment of all the atomic current loops is zero. • An external field alters electron motions within the atoms •  additional current loops • induced magnetic dipoles • The additional field is opposite to the direction of the external field (see Faraday’s law of induction next lecture!) • m < 0(in the range than 10−3 to 10−6) • Diamagnetic susceptibility is nearly temperature independent. • Examples of diamagnetic materials : glass and copper