Magnetic Materials — Formula Sheet

$m$: Magnetic dipole moment (A m²) · $B$: Uniform magnetic field (T) · $\theta$: Angle between m and B

Torque tends to align the dipole with the field. Maximum (mB) at θ = 90°, zero at θ = 0° or 180°. A uniform field gives no net force, only a torque. Unit: N m.

Minimum (−mB) when aligned with B (stable equilibrium). Maximum (+mB) when antiparallel (unstable equilibrium). Work done in rotating from θ₁ to θ₂: W = mB(cosθ₁ − cosθ₂). Unit: J.

$e$: Electronic charge = 1.6×10⁻¹⁹ C · $v$: Orbital speed of the electron (m s⁻¹) · $r$: Radius of the orbit (m) · $m_e$: Mass of electron = 9.1×10⁻³¹ kg · $L_{orb}$: Orbital angular momentum (kg m² s⁻¹ = J s)

Because the electron is negative, m_orb points opposite to its orbital angular momentum: m_orb = −(e/2mₑ)L_orb as vectors. The ratio e/2mₑ is the gyromagnetic ratio.

$h$: Planck's constant = 6.63×10⁻³⁴ J s

The smallest orbital magnetic moment of an electron (n = 1). Used as the natural unit of atomic magnetic moments. Equivalent unit: J T⁻¹.

$M$: Magnetisation (A m⁻¹) · $m_{net}$: Net magnetic dipole moment of the sample (A m²) · $V$: Volume of the sample (m³)

Net magnetic dipole moment per unit volume of the material. Unit: A m⁻¹ (same as H). It is a vector along the net dipole moment.

$H$: Magnetic intensity (A m⁻¹) · $B_0$: Applied magnetic field in vacuum (T) · ${\mu }_0$: Permeability of free space = 4π×10⁻⁷ T m A⁻¹ · $n$: Number of turns per unit length (m⁻¹)

Depends only on the free current, not on the material. B₀ is the field the current would produce in vacuum. Unit: A m⁻¹.

The field inside the material is the applied part μ₀H plus the contribution μ₀M of the material's own dipoles. Unit of B: tesla.

$\chi$: Magnetic susceptibility (no unit)

Dimensionless. Diamagnetic: small, negative. Paramagnetic: small, positive. Ferromagnetic: large, positive (and not constant — depends on H).

$\mu$: Permeability of the material (T m A⁻¹)

μ measures how easily a material lets magnetic field lines pass through it. Unit: T m A⁻¹ (= H m⁻¹).

${\mu }_r$: Relative permeability (no unit)

Dimensionless. μᵣ slightly less than 1 (diamagnetic), slightly more than 1 (paramagnetic), much greater than 1 (ferromagnetic).

Filling a solenoid or toroid with a material changes the field by the fraction χ. Multiply by 100 for percentage increase (paramagnetic) or decrease (diamagnetic).

$\Phi$: Magnetic flux (Wb) · $A$: Area of cross-section (m²)

Magnetic flux through the cross-section divided by area. Convert cm² to m² (1 cm² = 10⁻⁴ m²).

$C$: Curie constant (depends on the material) · $B$: Applied magnetic field, B = μ₀H (T) · $T$: Absolute temperature (K)

Magnetisation of a paramagnetic material is directly proportional to the applied field B and inversely proportional to the absolute temperature T. Putting B = μ₀H and χ = M/H gives χ = Cμ₀/T. T must be in kelvin. Valid for paramagnetic materials when the field is not too strong (far from saturation).

$N$: Number of atomic dipoles per unit volume (m⁻³) · $m_{atom}$: Dipole moment of each atom (A m²)

Maximum possible magnetisation, reached when every atomic dipole is aligned with the field. N is the number of atoms (dipoles) per unit volume.