Magnetism and Matter – Detailed Bullet-Point Notes

Introduction

  • Magnetism is pervasive: from galaxies to atoms, including Earth’s geomagnetic field that predates human evolution.
  • Word “magnet” comes from the Greek island Magnesia where natural magnetic ore (lodestone) was discovered (~600 BC).
  • 19th-century discoveries (Oersted, Ampère, Biot & Savart) linked moving charges/electric currents to magnetic fields.
  • Everyday observations:
    • Earth acts like a huge bar magnet: field lines emerge near the geographic south and enter near the geographic north.
    • A freely suspended bar magnet points N-S; north-seeking end → “north pole”, south-seeking end → “south pole”.
    • Like poles repel; unlike poles attract.
    • Isolating a single north or south pole is impossible; breaking a magnet yields two weaker magnets (no magnetic monopoles so far).
    • Iron and certain alloys can be magnetised.
  • Present chapter treats magnetism independently: bar magnet behaviour, Gauss’s law for magnetism, classification of materials (dia-, para-, ferromagnetism).

Bar Magnet

  • Sprinkling iron filings on glass above a bar magnet reveals a symmetric pattern indicating two poles and dipole nature.
  • Similar pattern forms around a current-carrying solenoid → visual link between magnets and solenoids.

Magnetic Field Lines

  • Imaginary lines that represent magnetic field $\mathbf{B}$.
  • Properties:
    • Form continuous closed loops (unlike electrostatic lines which start/end on charges).
    • Tangent at any point gives direction of $\mathbf{B}$.
    • Density (# lines per area) ∝ $|\mathbf{B}|$ (region ii in fig. shows stronger field than region i).
    • Lines never intersect; uniqueness of $\mathbf{B}$ would be lost.
  • Practical plotting: move a tiny compass and mark successive orientations.

Bar Magnet ≈ Equivalent Solenoid

  • Field patterns of a finite solenoid and bar magnet nearly identical at external points.
  • Thought-experiment: cut a solenoid; each piece retains continuous field loops, analogous to cutting a magnet.
  • Far-field (axial) of a finite solenoid and bar magnet:
    B<em>0=μ</em>04π2mr3B<em>0 = \frac{\mu</em>0}{4\pi}\frac{2m}{r^3}
    where $m$ = magnetic dipole moment, $r$ ≫ length $l$ of magnet.

Magnetic Dipole in Uniform Field

  • Place small magnet/needle (dipole moment $\mathbf{m}$) in uniform $\mathbf{B}$.
  • Torque:
    τ=m×B⇒∣τ∣=mBsin⁡θ\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B} \quad\Rightarrow\quad |\tau| = mB\sin\theta
    (restoring torque for small oscillations).
  • Magnetic potential energy relative to $\theta=90^\circ$: Um=−m⋅B=−mBcos⁡θU_m = -\mathbf{m}\cdot\mathbf{B}= -mB\cos\theta
    • Minimum (stable): $\theta=0^\circ$ ($U_{\min}=-mB$).
    • Maximum (unstable): $\theta=180^\circ$ ($U_{\max}=+mB$).

Electrostatic Analogy

  • Replace $(\mathbf{E},\,\mathbf{p},\,\tfrac{1}{4\pi\varepsilon0})$ with $(\mathbf{B},\,\mathbf{m},\,\tfrac{\mu0}{4\pi})$.
  • Equatorial field (for $r\gg l$): B<em>E=−μ</em>04πmr3B<em>E = -\frac{\mu</em>0}{4\pi}\frac{m}{r^{3}}
  • Axial field: B<em>A=μ</em>04π2mr3B<em>A = \frac{\mu</em>0}{4\pi}\frac{2m}{r^{3}}
  • Summary (Dipole Analogy Table):
    • Torque: $\mathbf{p}\times\mathbf{E}$ ⇔ $\mathbf{m}\times\mathbf{B}$
    • Energy: $-\mathbf{p}\cdot\mathbf{E}$ ⇔ $-\mathbf{m}\cdot\mathbf{B}$

Gauss’s Law for Magnetism

  • Magnetic lines form closed loops → net magnetic flux through any closed surface is zero.
  • Mathematically:
    ∮SB⋅dS=0\oint_{S}\mathbf{B}\cdot d\mathbf{S}=0
  • Contrast with electrostatics where ∮E⋅dS=q<em>encε</em>0\oint\mathbf{E}\cdot d\mathbf{S}=\frac{q<em>{\text{enc}}}{\varepsilon</em>0} because isolated charges (sources/sinks) exist.
  • Magnetic monopoles (isolated poles) have never been observed; if discovered, Gauss’s law would modify to ∮B⋅dS=μ<em>0q</em>m\oint\mathbf{B}\cdot d\mathbf{S}=\mu<em>0 q</em>m where $q_m$ = magnetic charge.

Magnetisation & Magnetic Intensity

  • For a material of volume $V$ with net magnetic moment $m{\text{net}}$: M=m</em>netV\mathbf{M}=\frac{m</em>{\text{net}}}{V} (units A m⁻¹).
  • Long solenoid (turn density $n$, current $I$) without core: B<em>0=μ</em>0nIB<em>0=\mu</em>0 n I
  • With magnetised core: B=B<em>0+B</em>mB = B<em>0 + B</em>m where B<em>m=μ</em>0 MB<em>m = \mu</em>0\,\mathbf{M}
  • Define magnetic intensity (also magnetising field):
    H=Bμ<em>0−M\mathbf{H}=\frac{\mathbf{B}}{\mu<em>0}-\mathbf{M} so B=μ</em>0(H+M)\mathbf{B}=\mu</em>0(\mathbf{H}+\mathbf{M})
  • Linear (isotropic) materials: M=χH\mathbf{M}=\chi\mathbf{H}
    • $\chi$ = magnetic susceptibility (dimensionless).
  • Hence, B=μ<em>0(1+χ)H=μ</em>0μrH=μH\mathbf{B}=\mu<em>0(1+\chi)\mathbf{H}=\mu</em>0\mu_r\mathbf{H}=\mu\mathbf{H}
    • $\mu_r = 1+\chi$ = relative permeability.
    • $\mu = \mu0\mur$ = absolute permeability.

Classification of Magnetic Materials

PropertyDiamagneticParamagneticFerromagnetic
Susceptibility $\chi$$-1\le\chi<0$ (≈ –10⁻⁵)$0<\chi<\varepsilon$ (≈ +10⁻⁵)$\chi\gg1$
Relative permeability $\mu_r$$0\le\mu_r<1$$1<\mu_r<1+\varepsilon$$\mu_r\gg1$
Response to external $\mathbf{B}$Repelled (move to weaker field)Weakly attracted (move to stronger field)Strongly attracted; can retain magnetisation

Diamagnetism

  • Resultant atomic magnetic moment = 0 in absence of field.
  • External $\mathbf{B}$ induces opposite magnetic moments (Lenz’s law) → net $\mathbf{M}$ opposes $\mathbf{B}$.
  • Field lines expelled slightly; inside field reduced.
  • Typical materials: Bi, Cu, Ag, Pb, Si, NaCl, water, N₂ (STP).
  • Superconductors: perfect diamagnets ($\chi = -1$, $\mu_r=0$, Meissner effect). Applications: magnetic levitation trains, MRI magnets.

Paramagnetism

  • Atoms/ions have permanent dipole moments; thermal agitation randomises them → no bulk $\mathbf{M}$.
  • External $\mathbf{B}$ partially aligns dipoles (alignment improves with stronger $\mathbf{B}$ or lower $T$) → small positive $\chi$.
  • Field inside slightly enhanced.
  • Materials: Al, Na, Ca, O₂ (STP), CuCl₂.
  • Shows saturation when all dipoles aligned.

Ferromagnetism

  • Strong, cooperative alignment: atoms group into “domains” (~1 mm, ~10¹¹ atoms) each with same orientation.
  • Without external field, domains random ⇒ net $\mathbf{M}=0$.
  • Applied $\mathbf{B}$ causes:
    • Domain walls shift; favourably oriented domains grow.
    • Eventually a single giant domain forms (magnet saturation).
  • Removal of field:
    • Hard ferromagnets (Alnico, lodestone) retain magnetisation → permanent magnets.
    • Soft ferromagnets (soft iron) lose magnetisation → useful for transformer cores.
  • Materials: Fe, Co, Ni, Gd, certain alloys; $\mu_r>1000$.
  • Above Curie temperature, ferromagnets become paramagnetic (thermal disruption of domains).

Worked-Out Example Highlights

  • Cutting a magnet (transverse or longitudinal) yields two complete dipoles; no isolated poles.
  • Magnetic needle in uniform field: torque exists but no net force; in non-uniform field, additional force arises.
  • Toroid produces internal $\mathbf{B}$ but net dipole moment = 0 → possible magnetic configuration without external poles.
  • Identifying magnetised vs unmagnetised iron bars: look for repulsion or field-strength variation (stronger at poles than centre).
  • Solenoid with magnetic core (given $n$, $I$, $\mu_r$): can compute $H$, $B$, $M$, and required magnetising current to mimic core effect.

Key Formula Compendium

  • Dipole torque: τ=mBsin⁡θ\tau = mB\sin\theta
  • Dipole energy: U=−mBcos⁡θU = -mB\cos\theta
  • Axial field ($r\gg l$): B<em>A=μ</em>04π2mr3B<em>A = \frac{\mu</em>0}{4\pi}\frac{2m}{r^3}
  • Equatorial field: B<em>E=−μ</em>04πmr3B<em>E = -\frac{\mu</em>0}{4\pi}\frac{m}{r^3}
  • Gauss (magnetism): ∮B⋅dS=0\oint\mathbf{B}\cdot d\mathbf{S}=0
  • Magnetisation: M=mnetV\mathbf{M}=\frac{m_{\text{net}}}{V}
  • Magnetising field: H=Bμ0−M\mathbf{H}=\frac{\mathbf{B}}{\mu_0}-\mathbf{M}
  • Linear medium: M=χH,B=μ0(1+χ)H=μH\mathbf{M}=\chi\mathbf{H},\quad \mathbf{B}=\mu_0(1+\chi)\mathbf{H}=\mu\mathbf{H}

Constants & Units (SI)

  • Permeability of free space: μ0=4π×10−7  N A−2\mu_0 = 4\pi\times10^{-7}\;\text{N A}^{-2}
  • 1 tesla (T) = 10⁴ gauss (G).
  • Magnetic moment unit: A m².
  • Magnetic flux: weber (Wb), 1 Wb=1 T⋅m21\,\text{Wb}=1\,\text{T·m}^2.

Summary Bullet List

  • Magnetic poles inseparable; simplest magnetic entity is a dipole.
  • Field lines: continuous, closed loops; density indicates strength; never cross.
  • Dipole experiences zero net force but torque $\mathbf{m}\times\mathbf{B}$ in uniform $\mathbf{B}$; potential energy $-\mathbf{m}\cdot\mathbf{B}$.
  • Far-field of bar magnet same as that of tiny current loop/solenoid.
  • Gauss’s law: zero net magnetic flux through any closed surface.
  • Magnetisation $\mathbf{M}$, magnetising field $\mathbf{H}$, susceptibility $\chi$, permeability $\mu$ inter-related via B=μH\mathbf{B}=\mu\mathbf{H}.
  • Diamagnetism (small negative $\chi$), Paramagnetism (small positive $\chi$), Ferromagnetism (large positive $\chi$, domain structure).
  • Ferromagnetism disappears above Curie temperature; soft vs hard ferromagnets determine temporary vs permanent magnets.
  • Superconductors show perfect diamagnetism (Meissner effect).

Points to Ponder (Condensed)

  • Engineering often precedes full scientific theory (e.g., compass before electromagnetism theory).
  • Charge is quantised; magnetic monopoles absent—mystery remains.
  • Perfect diamagnetism in superconductors links to perfect conductivity (BCS theory).
  • Tiny differences in $\chi$ create distinct macroscopic behaviours.
  • Beyond dia/para/ferro, exotic categories (antiferro-, ferri-, spin-glass) exist.