What Is Magnetic Permeability?
Magnetic permeability describes a material's ability to support magnetic field development within itself. It quantifies the relationship between the total magnetic field B and the applied external field H. Every substance falls into one of three categories:
- Diamagnetic materials possess permeability slightly below that of free space (μ₀). When exposed to an external field, they generate an internal field in the opposite direction, creating a weak repulsive effect. Most non-magnetic everyday materials—copper, water, and organic compounds—exhibit this behaviour.
- Paramagnetic materials have permeability slightly above μ₀. Their atoms contain unpaired electrons that weakly align with an external field, but this alignment vanishes once the field is removed. Aluminium and platinum are common examples.
- Ferromagnetic materials show dramatically higher permeability, often thousands of times greater than free space. Iron, cobalt, and nickel demonstrate strong, permanent magnetisation due to aligned electron spins throughout their crystalline structure.
The permeability of free space, μ₀, equals 4π × 10⁻⁷ H/m and serves as the reference baseline for all magnetic materials.
Permeability and Susceptibility Relationships
Three electromagnetic quantities describe a material's magnetic response. They are mathematically linked through simple proportionality constants:
μᵣ = μ / (4π × 10⁻⁷)
χ = μ / (4π × 10⁻⁷) − 1
χ = μᵣ − 1
μ— Absolute permeability in H/m (henries per metre)μᵣ— Relative permeability (dimensionless ratio compared to free space)χ— Magnetic susceptibility (dimensionless measure of magnetisation response)μ₀— Permeability of free space: 4π × 10⁻⁷ H/m
The Three Magnetic Fields
Understanding permeability requires distinguishing between three overlapping field quantities:
- Applied field H: The external magnetic field imposed on a material, measured in A/m. This originates from current-carrying coils or permanent magnets.
- Magnetisation M: The material's internal magnetic response, expressed as dipole moment per unit volume. It depends on the applied field and the material's susceptibility: M = χH.
- Total field B: The net magnetic field inside the material, combining the applied field and the material's response: B = μ₀(H + M) = μH.
In ferromagnetic materials, the relationship between B and H is non-linear and exhibits hysteresis—the magnetisation path depends on the field history, not just its current value.
Superconductors: Zero Permeability in Action
When certain materials are cooled below their critical temperature, they undergo a remarkable phase transition. Their electrical resistance drops to precisely zero, and their permeability becomes exactly zero as well. These superconductors are perfect diamagnets that expel all magnetic field lines from their interior—a phenomenon called the Meissner effect.
A superconductor levitates when placed on a magnet because the expelled field creates an equal and opposite force. The permeability of zero distinguishes a superconductor from a mere conductor: even a perfect conductor would allow existing fields to remain trapped, whereas a superconductor actively ejects them. Current applications include MRI machines, particle accelerators, and magnetic levitation systems. Practical superconductors require cooling with liquid nitrogen (77 K) or liquid helium (4 K), making their use specialised but increasingly valuable in high-energy physics and medical diagnostics.
Common Mistakes in Permeability Calculations
Permeability problems often stem from confusion between absolute and relative quantities, or mishandling of the reference constant.
- Forgetting the μ₀ reference — Relative permeability is dimensionless precisely because it divides absolute permeability by μ₀ = 4π × 10⁻⁷ H/m. Skipping this conversion produces an incorrect relative value. Always verify whether your result should be ~1, ~1000, or ~0.9 depending on material type.
- Confusing susceptibility with permeability — Susceptibility is permeability minus 1 (in relative units). For diamagnets, χ is slightly negative (−0.0001 range); for paramagnets, χ is small and positive; for ferromagnets, χ can be millions. They measure the same phenomenon but on different scales.
- Assuming linearity in ferromagnets — The relationship between B and H in iron and steel is highly non-linear. Permeability changes with applied field strength and magnetisation history. Simple proportionality formulas fail; hysteresis loops and magnetisation curves are required for accurate design.
- Ignoring temperature dependence — Permeability varies with temperature. Paramagnetic susceptibility follows Curie's law (decreasing with temperature). Ferromagnetic materials lose their properties above the Curie temperature, transitioning to paramagnetism. Always specify temperature when reporting permeability data.