Relative permeability

(PD) Image: John R. Brews
Infinitesimal current elements in two closed current-carrying loops

In physics, in particular in magnetostatics, the relative permeability is an intrinsic property of a magnetic material. It is usually denoted by μ_r. For simple magnetic materials, using SI units, μ_r is related to the proportionality constant between the magnetic flux density B and the magnetic field H, namely B = μ_r μ₀ H, where μ₀ is the magnetic constant. The relative permeability describes the ease by which a magnetic medium may be magnetized.

A related quantity is the magnetic susceptibility, denoted by χ_m, related to the magnetic permeability in SI units by:

$\mu _{r}=1+\chi _{m}\ .\ \mathrm {SI\ units} \ .$

The Ampère force exerted between infinitesimal current elements from two loops carrying currents I_i and I_j and immersed in a magnetic medium with relative permeability μ_r is changed by a factor μ_r. The infinitesimal current elements have vector components I_idℓ_i and I_jdℓ_j where the incremental lengths are directed along the wire at the location of the element, and pointing in the direction of the current. Ampère's original force law is:

$d^{2}\mathbf {F_{ij}} =-d^{2}\mathbf {F_{ji}} =I_{i}I_{j}{\frac {\mu _{r}\mu _{0}}{4\pi }}{\frac {\mathbf {{\hat {u}}_{ij}} }{r_{ij}^{2}}}\left(2\ (d{\boldsymbol {\ell _{i}\cdot }}d{\boldsymbol {\ell _{j}}})-3(\mathbf {{\hat {u}}_{ij}\cdot } d{\boldsymbol {\ell _{i}}})(\mathbf {{\hat {u}}_{ij}\cdot } d{\boldsymbol {\ell _{j}}})\right)\ ,$

with û_ij a unit vector pointing along the line joining element j to element i and r_ij the length of this line. The force element is second order because it is a product of two infinitesimals. This force law leads to the same force between closed current loops as the more commonly used Grassmann's law:

$d^{2}\mathbf {F_{ij}} =I_{i}I_{j}{\frac {\mu _{r}\mu _{0}}{4\pi }}{\frac {1}{r_{ij}^{2}}}\left(\ (d{\boldsymbol {\ell _{i}\cdot }}d{\boldsymbol {\ell _{j}}})\mathbf {{\hat {u}}_{ij}} -(\mathbf {{\hat {u}}_{ij}\cdot } d{\boldsymbol {\ell _{i}}})d{\boldsymbol {\ell _{j}}}\right)\ ,$

which is obviously not symmetric under exchange of the indices i and j, and violates Newton's law of action and reaction, but is readily derived from the Biot-Savart law, and is the expression for the force found in text books.

Empirically it is observed that the force may increase or decreases due to the presence of the magnetic medium, hence the relative permittivity μ_r may be greater than or less than 1. For simple media, if μ_r < 1, the medium is termed diamagnetic; if > 1 paramagnetic. Only classical vacuum has μ_r = 1 (exact). A typical diamagnetic susceptibility is about −10⁻⁵, while a typical paramagnetic susceptibility is about 10⁻⁴. It should be noted that the use of a constant as the relative permeability of a substance is an approximation, even for quantum vacuum. A more complete representation recognizes that all media exhibit departures from this approximation, in particular, a dependence on field strength, a dependence upon the rate of variation of the field in both time and space, and a dependence upon the direction of the field. In many materials these dependencies are slight; in others, like ferromagnets, they are pronounced.