Magnetism and Matter — Notes

• A freely suspended bar magnet points roughly north–south; its ends are the N and S poles. Like poles repel, unlike attract.
• Poles always occur in pairs: breaking a magnet gives two complete magnets. No magnetic monopoles have been found.
• Field lines are continuous closed loops: outside N → S, inside S → N. They never cross; crowding shows strength; the tangent gives the direction of $\vec B$.
• Because lines never start or end, the net magnetic flux through any closed surface is zero.
• Electric field lines of a dipole start on + and end on −; magnetic lines have no ends.

A current-carrying solenoid produces the same field pattern as a bar magnet (qualitative treatment). Its magnetic moment is $m = NIA$, directed along the axis towards the end where the current appears anticlockwise (the N end).

Inside a magnet, the orbital and spin motion of electrons in its atoms act like tiny aligned current loops; their moments add up to the magnet’s moment $\vec m$ (from S to N, unit A m²).

Cutting a solenoid in half gives two shorter solenoids, each with N and S ends — the same as breaking a magnet.

Field (far away, by analogy with the electric dipole):

• axial point: along $\vec m$, $\propto \dfrac{2m}{r^3}$;
• equatorial point: opposite to $\vec m$, $\propto \dfrac{m}{r^3}$ — half the axial value at the same $r$.

In a uniform field: net force zero; torque $\vec\tau = \vec m \times \vec B$ ($\tau = mB\sin\theta$) turns the magnet into line with $\vec B$. Stable equilibrium when $\vec m \parallel \vec B$ (energy $-mB$), unstable when antiparallel ($+mB$). A slightly displaced magnet oscillates about the field direction — the basis of the compass needle.

• Magnetisation $M$: net magnetic moment per unit volume, A m$^{-1}$.
• Magnetic intensity $H$: set by free currents (e.g. $H = nI$ in a solenoid), A m$^{-1}$.
• $\vec B = \mu_0(\vec H + \vec M)$; for linear materials $\vec M = \chi\vec H$, so $\vec B = \mu_0\mu_r\vec H$ with $\mu_r = 1 + \chi$ and permeability $\mu = \mu_0\mu_r$.
• $\chi$ and $\mu_r$ have no units.

Worked result: an iron core with $\mu_r = 500$ in a solenoid with $H = 2000$ A m$^{-1}$ gives $B \approx 1.26$ T and $M \approx 1.0 \times 10^{6}$ A m$^{-1}$.

• Superconductors are perfect diamagnets ($\chi = -1$, Meissner effect).
• Domains: regions in which atomic moments are spontaneously aligned. In a field, favourably aligned domains grow and others rotate.
• Soft ferromagnets (soft iron) magnetise and demagnetise easily — electromagnet and transformer cores. Hard ferromagnets (steel, alnico) retain magnetisation — permanent magnets.

• Diamagnetism: nearly independent of temperature.
• Paramagnetism: Curie law $\chi = \dfrac{C}{T}$ ($T$ in kelvin): thermal agitation opposes alignment, so χ falls as T rises; $\chi T$ is constant.
• Ferromagnetism: above the Curie temperature $T_C$ the domain alignment is destroyed and the material becomes paramagnetic (Fe ≈ 1043 K, Ni ≈ 631 K, Co ≈ 1394 K). A heated magnet loses its magnetism.