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Chapter 04 · Part V · Electromagnetism
Revised on ____________________
Chapter 04 · Magnetism and Matter

Bar magnets, the Earth, and the three families of material

2 sheets
2 diagrams
26 results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

I

The magnetic dipole

NO MONOPOLE HAS EVER BEEN FOUND
I.Moment  m = qm · 2l points S → N inside the magnet
II.Axial field  B = μ04π 2 mr³
III.Equatorial field  B = μ04π mr³ half the axial value, reversed
IV.Torque  τ = m × B
V.Energy  U = − m·B minimum when aligned
VI.Oscillating magnet  T = 2π √( Im B ) the vibration magnetometer
Magnetic field lines close on themselves: ∮B·dA = 0. That single statement is why cutting a magnet in half never yields an isolated pole.
II

The Earth's field

THREE ELEMENTS FIX IT COMPLETELY
I.Horizontal component  BH = B cos δ
II.Vertical component  BV = B sin δ
III.Angle of dip  tan δ = BVBH 0° at the equator, 90° at the poles
IV.Declination  geographic north − magnetic north
V.Apparent dip  cot²δ = cot²δ1 + cot²δ2 two perpendicular planes
VI.Neutral point  the magnet's field cancels BH
horizontalBB_H = B cosδB_V = B sinδδ (dip)tanδ = B_V / B_H
III

Matter in a field

B IS THE TOTAL FIELD, H IS NOT
I.Magnetising field  H = n I set by the free current alone
II.Magnetisation  M = magnetic momentvolume
III.Relation  B = μ0 (H + M)
IV.Susceptibility  χ = MH , μr = 1 + χ
V.Diamagnetic  χ small and negative expelled; no permanent moment
VI.Paramagnetic  χ small and positive aligns only while the field is on
VII.Ferromagnetic  χ ≫ 1 domains, remanence, hysteresis
VIII.Force uses  B, never H B = μ0H only in vacuum
IV

Temperature and hysteresis

THE LOOP IS THE ENERGY BILL
I.Curie's law  χ = CT paramagnets
II.Curie–Weiss  χ = C′T − TC a ferromagnet above its Curie point
III.Above TC  a ferromagnet becomes paramagnetic
IV.Retentivity  B when H returns to zero
V.Coercivity  the reverse H that erases B
VI.Loop area  = energy lost per unit volume per cycle soft iron: thin loop; steel: fat
HBB_r−H_csaturation
V

Where the marks go

FOUR ERRORS, EVERY YEAR
Where marks are lost in this chapter
1Reading zero net force as nothing happening. In a uniform field a dipole feels no force but a real torque.
2Asking where a magnetic field line begins. It does not — inside the magnet it runs S → N and closes.
3Substituting H for B in a force or torque. Those take B; inserting a core changes B, never H.
4Using χ = C/T just above the Curie point. A ferromagnet obeys C′/(T − TC) there.