Permeability

Prof. Dr. Thorsten Jungmann (last updated 2025-09-04)

Magnetic permeability (German: magnetische Permeabilität) describes the property of a material to conduct a magnetic field. It is also called magnetic conductivity and is the magnetic counterpart of permittivity in the electric field. In electrical engineering, it largely determines the inductance of inductors and the efficiency of transformers.

Tensor quantity in a homogeneous field with homogeneous material → scalar quantity

This is a tensor quantity. In this script, however, it is treated by convention as a scalar quantity in the context of the homogeneous field and under the assumption of homogeneous material.

1 Quantity symbol and unit

The quantity symbol of permeability is the Greek lower-case letter $\mu$ (‘mu’).
Its unit is henry per metre or newton per ampere squared:

$$[\mu ]=H/m=N/{\mathrm{A}}^2$$

2 Practical and worked examples

The (absolute) permeability ($\mu$) is the product of the relative permeability ${\mu }_r$ of the material and the vacuum permeability ${\mu }_0$:

$$\mu ={\mu }_r\cdot {\mu }_0$$

Vacuum permeability (${\mu }_0$):

$${\mu }_0=4\pi \cdot 10^{-7}H/m\approx 1.257\cdot 10^{-6}H/m$$

Relative permeability (${\mu }_r$):
A dimensionless number that states how much better a material conducts magnetic fields than the vacuum.

2.1 Inductance of an inductor with an iron core

An iron core gives an air-core inductor a considerably higher inductance:

$$L={\mu }_0\cdot {\mu }_r\cdot \frac{N^2\cdot A}{l}$$

Example: $N=200$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$, $l=0.02\mathrm{m}$

Air-core inductor (${\mu }_r=1$): $L_1=0.251\mathrm{mH}$

With iron core (${\mu }_r=2000$): $L_2=\underline{502\mathrm{mH}}$

Magnetic energy is stored in the iron core. After the inductor is switched off, this energy is discharged back into the circuit, unless this is prevented, for example, by a freewheeling diode.

2.2 Magnetic flux density in the material

The magnetic flux density is related to the magnetic field strength via the permeability:

$$B=\mu \cdot H={\mu }_0\cdot {\mu }_r\cdot H$$

Example: $H=1000A/m$, iron core with ${\mu }_r=1500$

$$B=1.257\cdot 10^{-6}H/m\cdot 1500\cdot 1000A/m=\underline{1.89\mathrm{T}}$$

3 Typical orders of magnitude

• Vacuum, air: ${\mu }_r=1$
• Paramagnetic materials (aluminium): ${\mu }_r=1.00002$
• Diamagnetic materials (copper): ${\mu }_r=0.99999$
• Ferromagnetic materials: ${\mu }_r=100$ to $10000$
• Ferrites: ${\mu }_r=10$ to $1000$

Ferrites

Ferrites are ceramic, ferrimagnetic materials based on iron oxide (Fe₂O₃) with other metal oxides such as nickel, zinc or manganese oxide.

Important properties:

• High magnetic permeability (${\mu }_r=10$ to $1000$)
• High electrical resistance (in contrast to iron)
• Low eddy current losses at higher frequencies

Typical applications in electrical engineering:

• Transformer cores for switch-mode power supplies
• Inductors in EMI filters
• RF signal transformers
• Toroidal cores for current transformers

The great advantage over metallic iron: because of the high electrical resistance, only small eddy currents arise in alternating magnetic fields, which reduces the losses. This is why ferrites are used particularly often at higher frequencies (kHz to MHz range).

Example: A switch-mode power supply transformer operates at 100 kHz – here an iron core would be unsuitable because of the eddy current losses, whereas a ferrite core is ideal.

4 Example phrasings

• ‘The iron core has a relative permeability of ${\mu }_r=2000$.’
• ‘The ferromagnetic material increases the permeability by a factor of 1500.’
• ‘The vacuum permeability is ${\mu }_0\approx 1.257\cdot 10^{-6}H/m$.’

5 Related quantities

• Magnetic flux density and magnetic field strength:

$$B=\mu \cdot H$$

• Inductance of an inductor:

$$L=\mu \cdot \frac{N^2\cdot A}{l}$$

• Reluctance of a magnetic circuit:

$$R_m=\frac{l}{\mu \cdot A}$$

• Magnetic flux:

$$\Phi =\frac{\Theta }{R_m}$$

6 Related components

• Inductor: the core material determines the inductance via the permeability
• Transformer: the iron core increases the efficiency through its high permeability
• Relay: a ferromagnetic armature enables a strong force effect