Magnetic materials

We have seen that the traditional idea of a ‘magnet’ actually corresponds to a magnetic moment, aka a magnetic dipole. Physically, this can be created by a loop of current. But inside a magnet, where do these current loops come from, and how are they sustained?

  • Even in a metal the magnetic moment does not come from movement of the conduction electrons. A current in this sense would encounter resistance and quickly die away1. And as we have seen, magnetic fields from typical currents make small forces, unlike those we experience from everyday magnets.
  • Instead the magnetic moment comes from the orbital motion of the electrons around the nucleus.
  • Because the electron orbits tight into the potential well where the Coulomb force is large, the centripetal acceleration is large and its speed is a substantial fraction ($\sim 1\,\%$) of the speed of light.
  • Quantum mechanics gives us that the orbital angular momentum comes in integer multiples of $\hbar \simeq 1.054 \times 10^{-34} {\rm kg\,m^2\,s^{-1}}$, and as a result the magnetic moment comes in integer multiples of $\mu_B \equiv e \hbar / (2 m_e) \simeq 9.274 \times 10^{-24}\,{\rm A\,m^2}$, known as the Bohr magneton.
  • There is also a contribution from the intrinsic ‘spin’ of the electrons, a quantum mechanical effect that loosely means even a single electron acts like it is spinning around its own axis. This turns out to come almost in multiples of $\mu_B$ too (don’t worry about the details of it for this year).
  • Due to this quantization, orbital and spin angular momentum cannot die away in the same way as a conventional current.
  • In the absence of an external magnetic field, a material can have zero magnetic moment if either:
    • the individual atoms have orbital+spin angular momentum that vector sums to zero (zero net magnetic moment per atom); or
    • the individual atoms have non-zero orbital+spin angular momentum, but it is randomly aligned and therefore has no large-scale effect (i.e. the vector sum over lots of atoms is roughly zero).

With this in mind, the most ubiquitous form of magnetism is diamagnetism, which is present in all materials:

  • This very weak form of magnetism arises because the orbit of the electrons is deformed by the presence of an external magnetic field (it now feels the force from $\vec{B}$ as well as from the Coulomb force of the nucleus).
  • It happens even if the atom doesn’t start out with a net magnetic moment; the orbits of the electrons are deformed so that a slight magnetic moment is induced (rather like a conductor can have an electric dipole induced once an external electric field is applied).
  • The deformation of the orbit generates a magnetic moment that opposes the external magnetic field, so the material is repelled by the field.2
  • Diamagnetism is always present (in every material), but also very weak, and easily overwhelmed by other magnetic effects.

Paramagnetism is a stronger form of magnetism:

  • This only occurs in materials with ‘unfilled shells’ – a technical way to say that each atom already has a net magnetic moment before the external field is applied.
  • In the absence of an external field, the magnetic moments are randomly oriented, so the material as a whole is not magnetic even though the individual atoms are.
  • When an external field is applied, the magnetic moments tend to align with the field, combining into a large-scale effect so the material is attracted to the field.
  • This is rather like how the electric dipoles align in a dielectric medium when exposed to an electric field (see lecture 8).

Ferromagnetism is the type of magnetism that we think of when we think of magnets:

  • This occurs in materials with unpaired electrons (like paramagnetism) but where, additionally, the magnetic moments of the atoms tend to align with each other.
  • This alignment can be maintained even after the external field is removed, so the material retains its magnetism. Permanent magnets are made out of ferromagnetic materials in this way.
  • In fact most paramagnetic materials become ferromagnetic at low temperatures. Whether or not a given material is ferromagnetic at room temperature comes down to a balance between thermal and magnetic energy, which in turn depends on the atomic and crystalline structure.

Permeability

For paramagnetic or diamagnetic materials, the field contributed by the material is directly proportional to the external field. Just as with dielectrics we defined a dielectric constant/relative permittivity $\kappa$ (see Lecture 8), we define a ‘relative permeability’ $K_m$:

\[K_m \equiv \frac{\vert \vec{B}_{\rm material} \vert}{\vert \vec{B}_{\rm vacuum}\vert},\]

which is the ratio between the actual magnetic field with the material present, and what it would have been if the material is replaced by a vacuum.

In laws like Ampère’s and Biot-Savart’s, we can replace $\mu_0 \to \mu$ where $\mu \equiv K_m \mu_0$, and we get the right answer for the magnetic field without having to explicitly model all the little magnetic dipoles inside the material.

Here are some typical relative permeabilities:

Material $K_m$
Vacuum 1
Air 1.0000004
Aluminium 1.000022
Water 0.999992
Superconductors 0
Steel 1000
Iron 5000

In the case of steel and iron, these are ferromagnetic materials so not only is the field vastly stronger than it would be in a vacuum, but it may also be maintained after the external field is removed. So, the permeability is not strictly a single constant of proportionality in these cases, and the values above are only indicative for cases starting from an unmagnetised state.

Discussion point: Why might the permeability of a superconductor be zero?

Solenoids

A ‘solenoid’ or ‘coil’ is a wire wound in a tight helix; it acts like lots of loops (or ‘turns’) stacked very nearly on top of each other. These turn out to be very useful in electrical engineering, for generating magnetic fields and for other purposes that we will see later.

  • If the solenoid is ‘short’, i.e. the length is much shorter than the radius, the magnetic fields from each loop superposes.
  • To get the total magnetic field, one therefore just multiplies the result for a single loop (from Y&F sec 28.5 / lecture 10) by $N$, where $N$ is the total number of loops.
  • The magnetic field in the centre is therefore $B = \mu_0 N I \hat{\vec{i}} / (2 a)$ where $a$ is the radius of a single loop, $I$ is the current flowing and the direction $\hat{\vec{i}}$ is normal to the plane of the loop, using the right-hand-rule.

However, it is also possible that the solenoid is ‘long’, i.e. the length of the whole solenoid is much larger than its radius.

  • Superposition is still valid, and you’ll use it to calculate the magnetic field in homework 7.
  • However, example 28.9 in YF shows how to calculate the magnetic field a different way – starting from Ampère’s law. We’ll sketch this out in the lecture.
  • Key to getting the Ampère derivation right is to set up a good path which passes through a few loops of the solenoid, then goes out to a very large distance before returning to the start.
  • Via either derivation, the magnetic field is given by $\vec{B} = \mu_0 n I \hat{\vec{i}}$, where $n$ is the number of loops per unit length, and $\hat{\vec{i}}$ is the direction of the solenoid taking the right-hand rule for the direction of the winding.

Discussion point: To get this result, YF just assert that the magnetic field outside the solenoid is zero. This can’t quite be true — the field lines have to loop back around the solenoid. Can you think of a good way to argue that the field outside the solenoid can still be neglected in the Ampère’s law calculation?

Footnotes

  1. The exception is in superconductors, where large-scale currents really can flow without any resistance at all. Superconductors therefore make excellent electromagnets. ↩

  2. In 1997, this was demonstrated by levitating a frog and a grasshopper – honestly, I am not sure why. But the person who did it, Andre Geim, also later won the Nobel Prize in physics for graphene. ↩