5. Magnetism And Matter

Magnets reveal both geometry and material behaviour. First model a small magnet as a dipole, then ask how many microscopic moments combine to give a sample’s macroscopic response.

The current-loop model connected permanent magnets with electric currents. Later studies of atomic moments and magnetic domains explained why some materials retain magnetisation while others respond only weakly to an applied field.

A permanent magnet and a current loop can produce similar external field patterns.

Matter contains microscopic sources of magnetic moment. Some materials respond weakly to an applied field; others can remain strongly magnetised after it is removed. This chapter connects the dipole picture to those different responses.

A freely turning compass aligns approximately with the local magnetic field. Its north-seeking end points along the field. The words north and south label magnetic poles; they do not describe separate magnetic charges that can be isolated by cutting an ordinary magnet.

A bar magnet is cut across its middle. What happens to its poles?

Each piece still contains microscopic magnetic moments. Cutting changes its dimensions, not the basic dipole character of the material.

Result: Each piece has both a north and a south pole; isolated poles are not produced.

1. What direction does the north end of a compass indicate?

Answers

1. The local magnetic-field direction, subject to mechanical and environmental disturbances.

A bar magnet is a magnetic dipole with a moment directed from its south end towards its north end.

Outside a bar magnet, field lines leave its north end and enter its south end. Inside the magnet, they continue from south towards north, forming closed loops. A tangent gives the local B direction. Dense line spacing indicates a strong field in a consistent sketch. Lines never cross where the field direction is well defined.

A solenoid with the same external dipole moment can model a short bar magnet at distances large compared with its length. A coil has $m=NIA$ in A m². This equivalence explains the similar field pattern without requiring actual isolated magnetic pole charges.

Far from a small magnetic dipole, the axial field is $B_{\rm ax}=\mu_0(2m)/(4\pi r^3)$ along m, while the equatorial field has magnitude $B_{\rm eq}=\mu_0m/(4\pi r^3)$ opposite m. These are far-field expressions. They are not accurate near the ends of a real bar magnet.

A small bar magnet has moment 0.80 A m². Estimate B at an axial distance of 0.20 m, assumed much greater than its size.

Use $B_{\rm ax}=2\times10^{-7}m/r^3$ with approximate μ₀.

Result: 20 μT along the moment.

In a uniform field, $\boldsymbol\tau=\mathbf m\times\mathbf B$ and $U=-\mathbf m\cdot\mathbf B$. Parallel alignment is stable. If a small magnet of rotational inertia Irot is displaced by a small angle θ, $I_{\rm rot}\ddot\theta=-mB\theta$. Its small-oscillation period is $T=2\pi\sqrt{I_{\rm rot}/(mB)}$, neglecting damping and other torques.

A magnet has m=0.20 A m² and rotational inertia 2.0 × 10⁻⁶ kg m² in B=50 μT. Estimate its small-oscillation period.

Use the small-angle approximation and express B in tesla.

Result: 2.8099 s.

Electric and magnetic dipoles have analogous far-field shapes, torque formulas and orientation energies. The key difference is that electric field lines may begin or end on electric charges, whereas magnetic field lines have no observed isolated source or sink. The analogy is useful only when that difference is retained.

1. At equal large r, how do axial and equatorial field magnitudes compare?

Answers

1. The axial value is twice the equatorial value.

A closed surface has zero net magnetic flux.

Gauss’s law for magnetism is $\displaystyle\oint\mathbf B\cdot d\mathbf A=0$. With outward area normals, every net inward contribution balances an outward contribution. The law expresses the absence of observed isolated magnetic monopoles in this classical description.

A Gaussian surface enclosing only the north end of a bar magnet still has zero net magnetic flux. Lines that leave through the surrounding space return through the magnet across the surface. The enclosed region is not a region containing an isolated magnetic charge.

The law does not imply zero B everywhere. A uniform nonzero field has equal inward and outward flux through opposite faces of a box. It also does not say that flux through an open surface is zero; a coil can have nonzero linked magnetic flux.

An imaginary closed box has +0.004 Wb of flux through five faces combined. What is the flux through the sixth?

Total outward magnetic flux must be zero.

Result: -0.004 Wb, inward.

1. Does enclosing a north pole without the south end make net magnetic flux positive?

Answers

1. No. Return field through the magnet completes the balance.

Magnetisation describes the magnetic moment density of matter.

The magnetisation vector is magnetic dipole moment per unit volume: $\mathbf M=\mathbf m_{\rm total}/\mathcal V$, in A/m. It describes a bulk response, not the magnetic moment of one atom. The magnetic field intensity H is related by $\mathbf B=\mu_0(\mathbf H+\mathbf M)$ in SI.

For a simple linear isotropic material, $\mathbf M=\chi_m\mathbf H$, giving $\mathbf B=\mu_0(1+\chi_m)\mathbf H=\mu\mathbf H$. Thus $\mu_r=1+\chi_m$ and μ=μ₀μr. Susceptibility χm is dimensionless. This linear relation is a model, not a universal law for a ferromagnet.

H helps separate the contribution associated with free currents from the material response. For a long uniformly filled solenoid with negligible end effects, H≈nI. In finite objects the sample shape can produce demagnetising fields, so H inside is not always the applied field that would exist without the sample.

A linear material has χm=0.020 and internal H=600 A/m. Find M and relative permeability.

Use $M=\chi_mH$ and $\mu_r=1+\chi_m$.

Result: M=12 A/m; μr=1.02.

A uniformly magnetised sample of volume 2.0 × 10⁻⁵ m³ has M=400 A/m. Find total moment.

Use $m=M\mathcal V$.

Result: 0.008 A m².

1. A linear material has negative χm. In what direction is its M relative to H?

Answers

1. Opposite.

Magnetic response depends on induced moments, permanent moments and collective ordering.

An applied field induces a weak response opposing the field. The susceptibility is small and negative; relative permeability is slightly below one. A diamagnetic sample is weakly pushed towards weaker-field regions in a nonuniform field. This induced response exists in ordinary matter, but stronger effects can mask it.

Atoms or ions with unpaired magnetic moments tend to align with an applied field, opposed by thermal agitation. The resulting susceptibility is usually small and positive. A sample is weakly drawn towards stronger-field regions. When the field is removed, ordinary paramagnetic materials lose the net alignment.

For many simple paramagnets in an appropriate temperature range, Curie’s law is $\chi_m=C/T$, with T in kelvin. It predicts a weaker alignment at a higher absolute temperature for the same field. It is not an exact relation for every magnetic material at every temperature.

In a ferromagnet, interactions favour collective alignment within domains. An unmagnetised specimen can have strongly magnetised domains whose moments cancel on a large scale. A field changes domain sizes and directions, creating a large net response. The relation between M and H is nonlinear and depends on magnetic history.

On removing the field, some magnetisation may remain: remanence. A reverse field is needed to reduce it to zero; the corresponding field scale is coercivity. A hysteresis loop records this history dependence. The loop area represents energy dissipated per cycle per volume when plotted with the appropriate conjugate magnetic quantities. Above a material’s Curie temperature, long-range ferromagnetic ordering disappears and its response becomes paramagnetic.

Soft magnetic materials are chosen for electromagnet cores that must magnetise and demagnetise easily, reducing hysteresis loss. Hard magnetic materials resist demagnetisation and suit permanent magnets. Large susceptibility alone is not enough to choose a material; remanence, coercivity, losses and working temperature matter.

A simple paramagnet has χm=0.004 at 300 K. Estimate χm at 450 K using Curie’s law.

The product $\chi_mT$ is constant in this model.

Result: 0.002667.

1. Why can an apparently unmagnetised ferromagnet contain magnetised regions?

Answers

1. Its domain moments may cancel macroscopically even though each domain is ordered.

A small magnetic dipole has torque $\mathbf m\times\mathbf B$ and energy $-\mathbf m\cdot\mathbf B$. Its far field scales as $r^{-3}$. Magnetic flux through a closed surface is zero.

Magnetisation is moment per volume. For a linear isotropic material, $\mathbf M=\chi_m\mathbf H$ and $\mu_r=1+\chi_m$. Diamagnetic, paramagnetic and ferromagnetic responses require different microscopic and macroscopic descriptions.

• Magnetic field lines close; cutting a magnet does not isolate a magnetic charge.
• B, H and M are related quantities with different units and roles.
• Ferromagnetic behaviour depends on history; a single linear susceptibility cannot describe the whole process.

Try these newly written questions before reading the answers. Show a diagram, the governing equation, substitutions and units where appropriate.

5.1 Cut a bar magnet into two pieces. What poles does each piece have?

5.2 State the external and internal directions of a bar magnet’s field lines.

5.3 What direction is assigned to the magnetic moment of a bar magnet?

5.4 A 50-turn coil carries 0.40 A and has area 0.003 m². Find its moment.

5.5 A short magnet has m=0.60 A m². Estimate axial B at r=0.30 m, assumed a far point.

5.6 Find its equatorial field magnitude at the same distance.

5.7 At fixed dipole moment, what happens to a far field when distance doubles?

5.8 A moment 0.30 A m² is at 90° to B=0.020 T. Find torque.

5.9 What is the energy difference between antiparallel and parallel orientations for that moment and field?

5.10 Which dipole orientation is stable in uniform B?

5.11 How does a small-oscillation period change if B increases by a factor of four?

5.12 What is total outward magnetic flux through a closed sphere surrounding a magnet?

5.13 Can an open surface have nonzero magnetic flux?

5.14 A sample has total moment 0.015 A m² and volume 5 × 10⁻⁵ m³. Find M.

5.15 A linear material has χm=0.005 and H=800 A/m. Find M and μr.

5.16 Which broad response has negative susceptibility?

5.17 Why does an ordinary paramagnet lose net magnetisation after removal of the field?

5.18 A simple paramagnet has χm=0.006 at 250 K. Estimate it at 500 K.

5.19 Why is the Curie-law temperature expressed in kelvin?

5.20 What is remanence?

5.21 What does coercivity describe?

5.22 Why choose a soft magnetic core for a repeatedly switched electromagnet?

5.23 Why can a permanent magnet remain magnetised without an external field?

5.24 What happens above a ferromagnet’s Curie temperature?

Answers

5.1 Each has both north and south poles.

5.2 Outside N to S; inside S to N, making closed loops.

5.3 From its south end towards its north end.

5.4 0.06 A m².

5.5 4.4444 μT.

5.6 2.2222 μT, opposite m.

5.7 Its magnitude divides by eight.

5.8 6 mN m.

5.9 12 mJ.

5.10 Parallel.

5.11 It halves for unchanged moment and inertia.

5.12 Zero.

5.13 Yes.

5.14 300 A/m.

5.15 M=4 A/m; μr=1.005.

5.16 Diamagnetism.

5.17 Thermal disorder removes the preferred alignment of its individual moments.

5.18 0.003 using Curie’s law.

5.19 The law uses absolute temperature; Celsius zero is not the relevant physical zero.

5.20 Magnetisation remaining after the applied field is reduced to zero along a specified history.

5.21 The reverse-field scale required to remove the residual magnetisation in the specified hysteresis measurement.

5.22 It can magnetise and demagnetise readily with small coercivity and reduced hysteresis loss.

5.23 Domain alignment can persist because of collective ordering and resistance to domain reversal.

5.24 Its long-range ferromagnetic order disappears and it becomes paramagnetic.