Lesson 11 of 11 · 15 min
Chapter review
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Must-know facts
18 facts
- 01Magnetic monopoles are not known to exist; a broken magnet gives two complete magnets.
- 02Field lines are closed loops, N to S outside and S to N inside; they never cross.
- 03A bar magnet behaves like a solenoid of the same moment m = NIA.
- 04In a uniform field a magnet feels torque mB sin θ but zero net force.
- 05U = −mB cos θ, zero at 90°; −mB at 0° (stable), +mB at 180° (unstable).
- 06Work to turn a magnet from along B to opposite B is 2mB.
- 07Far axial field (μ₀/4π)(2m/r³); equatorial (μ₀/4π)(m/r³), opposite to m.
- 08Replace E → B, p → m, 1/4πε₀ → μ₀/4π to get magnetic dipole results.
- 09Gauss's law for magnetism: net flux of B through any closed surface is zero.
- 10M = m_net/V, unit A m⁻¹; H = B/μ₀ − M, unit A m⁻¹; B = μ₀(H + M).
- 11M = χH; μᵣ = 1 + χ; μ = μ₀μᵣ.
- 12In a solenoid H = nI whatever the core.
- 13Diamagnetic: χ negative, μᵣ < 1, repelled; bismuth, copper, lead, water, NaCl, N₂.
- 14Paramagnetic: χ small positive, weakly attracted; Al, Na, Ca, O₂, CuCl₂; χ depends on temperature.
- 15Ferromagnetic: χ ≫ 1, μᵣ above 1000; Fe, Co, Ni, Gd; domains of about 1 mm, 10¹¹ atoms.
- 16Superconductor: χ = −1, μᵣ = 0, field fully expelled (Meissner effect).
- 17Hard ferromagnets (alnico, lodestone) make permanent magnets; soft iron loses its magnetism.
- 18A ferromagnet heated enough becomes paramagnetic.
Common traps
Where marks are lost
Formulas
9 to know
Torque on a magnet
τ = m × B, τ = mB sin θ
θ between m and B; zero net force in a uniform field.
Potential energy of a magnet
U = −m·B = −mB cos θ
Zero at θ = 90°; minimum −mB at 0°, maximum +mB at 180°.
Moment of an equivalent solenoid
m = NIA
Unit A m² = J/T.
Axial field of a short magnet
B_A = (μ₀/4π)(2m/r³)
r ≫ size of the magnet; along m.
Equatorial field of a short magnet
B_E = −(μ₀/4π)(m/r³)
Opposite to m; half the axial size.
Gauss's law for magnetism
Σ B·ΔS = 0 over any closed surface
No magnetic monopoles.
Magnetisation
M = m_net/V
Unit A m⁻¹.
Field in a material
B = μ₀(H + M), H = B/μ₀ − M
In a solenoid H = nI.
Susceptibility and permeability
M = χH, μᵣ = 1 + χ, μ = μ₀μᵣ, B = μH
χ and μᵣ are dimensionless; μ has the units of μ₀.
Key terms
17 terms
- Magnetic monopole
- An isolated north or south pole; none is known to exist.
- Magnetic field line
- A closed curve whose tangent at each point gives the direction of B; closer lines mean a stronger field.
- Magnetic moment (m)
- The vector that fixes a magnet's torque and far field; NIA for an equivalent solenoid, unit A m² or J/T.
- Magnetic potential energy
- U = −m·B for a dipole in a uniform field, taken as zero when m is at right angles to B.
- Equatorial line
- The normal bisector of a magnet, where its far field is half the axial value and opposite to m.
- Magnetic flux
- B·ΔS summed over a surface; unit weber (T m²).
- Gauss's law for magnetism
- The net magnetic flux out of any closed surface is zero.
- Magnetisation (M)
- Net magnetic moment per unit volume of a sample, in A m⁻¹.
- Magnetic intensity (H)
- B/μ₀ − M, the part of the field set by external currents; nI in a solenoid.
- Magnetic susceptibility (χ)
- The dimensionless ratio M/H, measuring how strongly a material responds to a field.
- Relative permeability (μᵣ)
- 1 + χ, the factor by which a material multiplies μ₀; the magnetic twin of the dielectric constant.
- Diamagnetic
- Having small negative χ; pushed from strong field to weak.
- Paramagnetic
- Having small positive χ from permanent atomic moments; weakly pulled into a strong field.
- Ferromagnetic
- Having χ ≫ 1 because atomic moments align in domains; strongly attracted.
- Domain
- A region, typically about 1 mm across with about 10¹¹ atoms, in which all atomic moments of a ferromagnet point one way.
- Meissner effect
- The complete expulsion of a magnetic field from a superconductor, which is a perfect diamagnet.
- Hard and soft ferromagnets
- Hard ones keep their magnetisation when the field is removed (alnico, lodestone); soft ones lose it (soft iron).