5.6 Summary

• The science of magnetism is old. It has been known since ancient times that magnetic materials tend to point in the north-south direction; like magnetic poles repel and unlike ones attract; and cutting a bar magnet in two leads to two smaller magnets. Magnetic poles cannot be isolated.
• When a bar magnet of dipole moment \(\mathbf{m}\) is placed in a uniform magnetic field \(\mathbf{B}\),
  (a) the force on it is zero,
  (b) the torque on it is \(\mathbf{m} \times \mathbf{B}\),
  (c) Its potential energy is \(\mathbf{- m} \cdot \mathbf{B}\), where we choose the zero of energy at the orientation when \(\mathbf{m}\) is perpendicular to \(\mathbf{B}\).
• Consider a bar magnet of size \(l\) and magnetic moment \(\mathbf{m}\), at a distance \(r\) from its mid-point, where \(r \gg l\), the magnetic field \(\mathbf{B}\) due to this bar is,
  \(
  \begin{aligned}
  \mathbf{B} & =\frac{\mu_0 \mathbf{m}}{2 \pi r^3} \\
  & \text { (along axis) } \\
  & =-\frac{\mu_0 \mathbf{m}}{4 \pi r^3} \quad \text { (along equator) }
  \end{aligned}
  \)
• Gauss’s law for magnetism states that the net magnetic flux through any closed surface is zero
  \(
  \phi_B=\sum_{\substack{\text { all area } \\ \text { elements } \Delta \mathbf{S}}} \mathbf{B} \cdot \Delta \mathbf{S}=0
  \)
• Consider a material placed in an external magnetic field \(\mathbf{B}_0\). The magnetic intensity is defined as,
  \(
  \mathbf{H}=\frac{\mathbf{B}_0}{\mu_0}
  \)
  The magnetisation \(\mathbf{M}\) of the material is its dipole moment per unit volume. The magnetic fleld \(\mathbf{B}\) in the material is,
  \(
  \mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})
  \)
• For a linear matertal \(\mathbf{M}=\chi \mathbf{H}\). So that \(\mathbf{B}=\mu \mathbf{H}\) and \(\chi\) is called the magnetic susceptibility of the material. The three quantities, \(\chi\), the relative magnetic permeability \(\mu_{\mathrm{r}}\), and the magnetic permeability \(\mu\) are related as follows:
  \(
  \begin{aligned}
  & \mu=\mu_0 \mu_r \\
  & \mu_r=1+\chi
  \end{aligned}
  \)
• Magnetic materials are broadly classified as diamagnetic, paramagnetic, and ferromagnetic. For diamagnetic materials \(\chi\) is negative and small, and for paramagnetic materials it is positive and small. Ferromagnetic materials have large \(\chi\) and are characterised by non-linear relation between \(\mathbf{B}\) and \(\mathbf{H}\).
• Substances, which at room temperature retain their ferromagnetic property for a long period of time, are called permanent magnets.

POINTS TO PONDER

• A satisfactory understanding of magnetic phenomenon in terms of moving charges/currents was arrived at after 1800 AD. But technological exploitation of the directional properties of magnets predates this scientific understanding by two thousand years. Thus, scientific understanding is not a necessary condition for engineering applications. Ideally, science and engineering go hand-in-hand, one leading and assisting the other in tandem.
• Magnetic monopoles do not exist. If you slice a magnet in half, you get two smaller magnets. On the other hand, isolated positive and negative charges exist. There exists a smallest unit of charge, for example, the electronic charge with value \(|e|=1.6 \times 10^{-19} \mathrm{C}\). All other charges are integral multiples of this smallest unit charge. In other words, charge is quantised. We do not know why magnetic monopoles do not exist or why electric charge is quantised.
• A consequence of the fact that magnetic monopoles do not exist is that the magnetic field lines are continuous and form closed loops. In contrast, the electrostatic lines of force begin on a positive charge and terminate on the negative charge (or fade out at infinity).
• A minuscule difference in the value of \(\chi\), the magnetic susceptibility, yields radically different behaviour: diamagnetic versus paramagnetic. For diamagnetic materials \(\chi=-10^{-5}\) whereas \(\chi=+10^{-5}\) for paramagnetic materials.
• There exists a perfect diamagnet, namely, a superconductor. This is a metal at very low temperatures. In this case \(\chi=-1, \mu_r=0, \mu=0\). The external magnetic field is totally expelled. Interestingly, this material is also a perfect conductor. However, there exists no classical theory which ties these two properties together. A quantum-mechanical theory by Bardeen, Cooper, and Schrieffer (BCS theory) explains these effects. The BCS theory was proposed in 1957 and was eventually recognised by a Nobel Prize in physics in 1970.
• Diamagnetism is universal. It is present in all materials. But it is weak and hard to detect if the substance is para- or ferromagnetic.
• We have classified materials as diamagnetic, paramagnetic, and ferromagnetic. However, there exist additional types of magnetic material such as ferrimagnetic, anti-ferromagnetic, spin glass, etc. with properties which are exotic and mysterious.