Magnetism and Matter – Comprehensive Bullet-Point Notes

Introduction to Magnetism

• Magnetic phenomena are universal, manifesting from galactic to atomic scales.
• Etymology: “Magnet” derives from Magnesia, an island in Greece known for magnetic ore (≈600 BC).
• Historical milestones (early 19th C): discovery that electric currents produce magnetic fields—credited to Ørsted, Ampère, Biot & Savart.

Fundamental Observations

• Earth behaves like a huge bar magnet; field points ≈ from geographic South to North.
• Freely suspended bar magnets align N–S; ends are termed north pole and south pole.
• Like poles repel; unlike poles attract.
• Magnetic monopoles (isolated N or S poles) do not exist—breaking a magnet yields two weaker magnets, each with N & S.
• Iron and specific alloys can be magnetised.

The Bar Magnet

Iron-filing experiment

• Sprinkling filings on a glass sheet above a bar magnet reveals a dipole-like pattern.
• Similar pattern appears around a current-carrying solenoid ⇒ analogy between magnets & current loops.

Magnetic Field Lines

• Visual, intuitive representation of $\mathbf B$.
• Properties:
  • Form continuous closed loops (contrast with electric dipole lines that begin/end on charges).
  • Tangent at any point gives direction of $\mathbf B$.
  • Density of lines ∝ magnitude of $|\mathbf B|$.
  • Never intersect; unique field direction at every point.
• Plotting: move a small compass to map orientations.

Bar Magnet ≈ Equivalent Solenoid

• A bar magnet can be modelled as many circulating microscopic currents.
• Cutting a bar magnet resembles cutting a solenoid—continues to show closed field lines.
• Axial field of a finite solenoid (for distant point $r\gg l$):
  $B<em>\text{axial}=\frac{\mu</em>0}{4\pi}\,\frac{2m}{r^{3}}$
  —identical to far-field of a magnetic dipole (bar magnet) of moment $m$.
• Magnetic moment of bar magnet equals that of the equivalent solenoid.

Magnetic Dipole in Uniform Field

• Torque: $\boldsymbol\tau = \mathbf m \times \mathbf B$; magnitude $\tau = mB\sin\theta$.
• Potential energy (zero taken at $\theta = 90^{\circ}$):
  $U_m = -\mathbf m\cdot\mathbf B = -mB\cos\theta$
  • Minimum $= -mB$ at $\theta = 0^{\circ}$ (stable).
  • Maximum $= +mB$ at $\theta = 180^{\circ}$ (unstable).

Electrostatic Analogy (large-distance fields)

• Replace $\mathbf E \to \mathbf B$, $\mathbf p\to\mathbf m$, $1/4\pi\varepsilon<em>0\to\mu</em>0/4\pi$.
• Equatorial field: $B<em>E = -\frac{\mu</em>0}{4\pi}\,\frac{m}{r^{3}}$.
• Axial field: $B<em>A = \frac{\mu</em>0}{4\pi}\,\frac{2m}{r^{3}}$.

Gauss’s Law for Magnetism

• For any closed surface $S$, net magnetic flux is zero:
  $\oint_S \mathbf B\cdot d\mathbf S = 0$.
• Reflects non-existence of magnetic monopoles; $\mathbf B$ lines form closed loops.
• Contrasts with electrostatics: $\oint<em>S \mathbf E\cdot d\mathbf S = q/\varepsilon</em>0$.

Magnetisation & Magnetic Intensity

• Magnetisation (vector): $\mathbf M = \dfrac{\text{net magnetic moment}}{\text{volume}}$; units A m⁻¹.
• For a long solenoid (turn density $n$, current $I$): field without core $B<em>0 = \mu</em>0 n I$.
• With magnetic core: $\mathbf B = \mu<em>0(\mathbf H + \mathbf M)$ where • Magnetic intensity $\mathbf H = \dfrac{\mathbf B}{\mu</em>0} - \mathbf M$.
• Linear (isotropic) materials: $\mathbf M = \chi \mathbf H$.
  • $\chi$ = magnetic susceptibility (dimensionless).
• Hence $\mathbf B = \mu<em>0(1+\chi)\mathbf H = \mu \mathbf H$. • Relative permeability $\mu</em>r = 1+\chi$.
  • Absolute permeability $\mu = \mu<em>0\mu</em>r$.

Worked Example (core in solenoid)

• Given $n=1000\,\text{m}^{-1}, I=2\,\text{A}, \mu<em>r=400$: • $H = nI = 2\times10^{3}\,\text{A m}^{-1}$. • $B = \mu</em>r\mu<em>0 H = 1.0\,\text{T}$. • $M = (\mu</em>r-1)H \approx 8\times10^{5}\,\text{A m}^{-1}$.
  • Magnetising current to replicate $B$ without core $I_m \approx 794\,\text{A}$.

Magnetic Properties of Materials

Diamagnetism

• Net atomic magnetic moment $=0$; applied field induces opposite moment via Lenz-like reaction.
• Field lines expelled (slightly); sample moves from strong → weak field.
• Examples: Bi, Cu, Pb, water, NaCl.
• Superconductors: perfect diamagnets (Meissner effect) with $\chi=-1, \mu_r=0$ ⇒ complete field expulsion; enable magnetic levitation.

Paramagnetism

• Atoms possess permanent dipole moment but random thermal agitation ⇒ zero net $\mathbf M$.
• External $\mathbf B_0$ partially aligns dipoles; internal field slightly enhanced.
• Sample drifts toward stronger field regions.
• $\chi,\,\mu_r$ depend on T; magnetisation saturates when alignment is complete.
• Examples: Al, Na, Ca, O₂, CuCl₂.

Ferromagnetism

• Strong, spontaneous alignment of dipoles within domains (≈1 mm, ≈10¹¹ atoms).
• Without field, domains randomly oriented ⇒ no bulk $\mathbf M$; field causes domains to grow/align producing large $\mathbf B$ concentration.
• Hard ferromagnets (Alnico, lodestone) retain $\mathbf M$ after field removal → permanent magnets.
• Soft ferromagnets (soft iron) lose $\mathbf M$ when field removed → used in transformer cores.
• Above Curie temperature, ferromagnets transition to paramagnetic.

Examples & Concept Checks

• Cutting a magnet (transverse or along length) yields two smaller dipoles—no monopoles.
• Magnet in uniform field experiences torque but zero net force; in non-uniform field there is force.
• Toroid field lines are closed within core, giving zero net dipole moment (no external poles).
• Distinguish magnetised vs un-magnetised bars via attraction pattern (repulsion confirms mutual magnetisation; mid-point test locates strongest pole regions).
• Magnetic field lines must neither start/stop in space nor intersect; diagrams violating these rules are incorrect.
• Gauss’s law would acquire $\mu<em>0 q</em>m$ term if monopoles $q_m$ existed.

Key Formula Compendium

• Torque on dipole: $\boldsymbol\tau = \mathbf m \times \mathbf B$.
• Potential energy: $U = -\mathbf m\cdot\mathbf B$.
• Axial field (dipole, $r\gg l$): $B<em>A = \dfrac{\mu</em>0}{4\pi}\dfrac{2m}{r^{3}}$.
• Equatorial field: $B<em>E = -\dfrac{\mu</em>0}{4\pi}\dfrac{m}{r^{3}}$.
• Magnetisation: $\mathbf M = \dfrac{\mathbf m_{\text{net}}}{V}$.
• Relation of fields: $\mathbf B = \mu_0(\mathbf H + \mathbf M)$; $\mathbf M = \chi \mathbf H$.
• Permeability links: $\mu = \mu<em>0(1+\chi)=\mu</em>0\mu_r$.

Connections & Implications

• Magnetic fields from moving charges unify electricity & magnetism; yet magnetic monopole absence distinguishes two domains.
• Quantum mechanics (e.g., BCS, exchange interaction) required to fully explain superconductivity & ferromagnetism.
• Technological leverage: permanent magnets, transformers (soft ferromagnets), MRI (superconducting coils), mag-lev trains (Meissner diamagnetism).

Ethical & Practical Notes

• Permanent magnets in everyday gadgets (speakers, sensors) require environmentally safe mining/processing of rare ferromagnetic elements.
• Superconducting technologies promise energy-efficient transport but necessitate cryogenic infrastructure with economic and ecological considerations.

Common Units & Dimensions (quick recall)

• $\mathbf B$: tesla (T) = N s C⁻¹ m⁻¹.
• Magnetic moment $m$: A m².
• Flux $\phi_B$: weber (Wb) = T m².
• $\mathbf H, \mathbf M$: A m⁻¹.
• Permeability $\mu, \mu_0$: T m A⁻¹.

Exam Tips

• Always sketch field lines to visualise torque/force situations.
• For dipole problems, decide quickly whether to use axial or equatorial formula.
• Remember sign conventions: $\mathbf M$ opposite $\mathbf H$ in diamagnets.
• When asked about stability, link to potential-energy minima.
• In mixed-concept questions (Gauss + magnetism), emphasise closed-loop nature of $\mathbf B$.