In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction B {\displaystyle B} to the magnetizing field H {\displaystyle H} in a material. The term was coined by Lord Kelvin in 1872, and is used alongside its electrostatic equivalent, permittivity, coined by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity. In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per square ampere (N/A2). The permeability constant μ0, also known as the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum. A closely related property of materials is magnetic susceptibility, which is a dimensionless proportionality factor that indicates the degree of magnetization of a material in response to an applied magnetic field.
Explanation In the macroscopic formulation of electromagnetism, there appear two different kinds of magnetic field:
the magnetizing field H which is generated around electric currents and displacement currents, and also emanates from the poles of magnets. The SI units of H are amperes per meter. the magnetic flux density B which acts back on the electrical domain, by curving the motion of charges and causing electromagnetic induction. The SI units of B are volt-seconds per square meter, a ratio equivalent to one tesla. The concept of permeability arises since in many materials (and in vacuum), there is a simple relationship between H and B at any location or time, in that the two fields are precisely proportional to each other:
B = μ H , {\displaystyle \mathbf {B} =\mu \mathbf {H} ,}
where the proportionality factor μ is the permeability, which depends on the material. The permeability of vacuum (also known as permeability of free space) is a physical constant, denoted μ0. The SI units of μ are volt-seconds per ampere-meter, equivalently henry per meter. Typically μ would be a scalar, but for an anisotropic material, μ could be a second rank tensor. However, inside strong magnetic materials (such as iron, or permanent magnets), there is typically no simple relationship between H and B. The concept of permeability is then nonsensical or at least only applicable to special cases such as unsaturated magnetic cores. Not only do these materials have nonlinear magnetic behaviour, but often there is significant magnetic hysteresis, so there is not even a single-valued functional relationship between B and H. However, considering starting at a given value of B and H and slightly changing the fields, it is still possible to define an incremental permeability as:
Δ B = μ Δ H . {\displaystyle \Delta \mathbf {B} =\mu \,\Delta \mathbf {H} .}
assuming B and H are parallel. In the microscopic formulation of electromagnetism, where there is no concept of an H field, the vacuum permeability μ0 appears directly (in the SI Maxwell's equations) as a factor that relates total electric currents and time-varying electric fields to the B field they generate. In order to represent the magnetic response of a linear material with permeability μ, this instead appears as a magnetization M that arises in response to the B field: M = ( μ 0 − 1 − μ − 1 ) B {\displaystyle \mathbf {M} =\left(\mu _{0}^{-1}-\mu ^{-1}\right)\mathbf {B} } . The magnetization in turn is a contribution to the total electric current—the magnetization current.
Relative permeability and magnetic susceptibility Relative permeability, denoted by the symbol μ r {\displaystyle \mu _{\mathrm {r} }} , is the ratio of the permeability of a specific medium to the permeability of free space μ0:
μ r = μ μ 0 , {\displaystyle \mu _{\mathrm {r} }={\frac {\mu }{\mu _{0}}},}
where μ 0 ≈ {\displaystyle \mu _{0}\approx } 4π × 10−7 H/m is the magnetic permeability of free space. In terms of relative permeability, the magnetic susceptibility is
χ m = μ r − 1. {\displaystyle \chi _{m}=\mu _{r}-1.}
The number χm is a dimensionless quantity, sometimes called volumetric or bulk susceptibility, to distinguish it from χp (magnetic mass or specific susceptibility) and χM (molar or molar mass susceptibility).
Diamagnetism
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