Lesson 11 of 11 · 15 min

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Must-know facts

18 facts

1Magnetic monopoles are not known to exist; a broken magnet gives two complete magnets.
2Field lines are closed loops, N to S outside and S to N inside; they never cross.
3A bar magnet behaves like a solenoid of the same moment m = NIA.
4In a uniform field a magnet feels torque mB sin θ but zero net force.
5U = −mB cos θ, zero at 90°; −mB at 0° (stable), +mB at 180° (unstable).
6Work to turn a magnet from along B to opposite B is 2mB.
7Far axial field (μ₀/4π)(2m/r³); equatorial (μ₀/4π)(m/r³), opposite to m.
8Replace E → B, p → m, 1/4πε₀ → μ₀/4π to get magnetic dipole results.
9Gauss'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

Taking the potential energy of a magnet to be zero when it lies along the field.

U = −mB cos θ is zero at 90°. Along B it is −mB, the minimum; opposite to B it is +mB.

Giving the work to turn a magnet from 0° to 180° as mB.

W = U(180°) − U(0°) = mB − (−mB) = 2mB. From 0° to 90° it is mB.

Thinking the equatorial field of a magnet points the same way as its moment.

B_E = −(μ₀/4π)m/r³: opposite to m, and half the axial value at the same distance.

Treating magnetic field lines as lines of force on a moving charge.

The force qv × B is perpendicular to B, so the lines only give the direction of B.

Saying field lines start at the N pole and end at the S pole.

They continue through the magnet from S to N and form closed loops; that is why the closed-surface flux is zero.

Expecting a uniform field to pull a magnet or a needle towards it.

A uniform field gives only a torque. A net force needs a non-uniform field, as near a pole.

Calling copper or water paramagnetic, or aluminium diamagnetic.

Copper, bismuth, lead, silicon, water, NaCl and N₂ are diamagnetic; aluminium, sodium, calcium, O₂ and CuCl₂ are paramagnetic.

Writing μᵣ = χ or μ = μ₀χ.

μᵣ = 1 + χ and μ = μ₀(1 + χ). For a diamagnet μᵣ is just below 1, not negative.

Thinking H in a solenoid changes when an iron core goes in.

H = nI is set by the winding; the core changes M and so B = μ₀(H + M).

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).

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