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What is susceptibility

  1. Jul 24, 2014 #1
    Definition/Summary

    Susceptibility is a property of material. In a vacuum it is zero.

    Susceptibility is an operator (generally a tensor), converting one vector field to another. It is dimensionless.

    Electric susceptibility [itex]\chi_e[/itex] is a measure of the ease of polarisation of a material.

    Magnetic susceptibility [itex]\chi_m[/itex] is a measure of the strengthening of a magnetic field in the presence of a material.

    Diamagnetic material has negative magnetic susceptibility, and so weakens a magnetic field.

    Equations

    Electric susceptibility [itex]\chi_e[/itex] and magnetic susceptibility [itex]\chi_m[/itex] are the operators which convert the electric field and the magnetic intensity field, [itex]\varepsilon_0\mathbf{E}[/itex] and [itex]\mathbf{H}[/itex] ([itex]not[/itex] the magnetic field [itex]\mathbf{B}[/itex]), respectively, to the polarisation and magnetisation fields [itex]\mathbf{P}[/itex] and [itex]\mathbf{M}[/itex]:

    [tex]\mathbf{P}\ = \chi_e\,\varepsilon_0\,\mathbf{E}[/tex]

    [tex]\mathbf{M}\ = \chi_m\,\mathbf{H}\ = \frac{1}{\mu_0}\,\chi_m\,(\chi_m\,+\,1)^{-1}\,\mathbf{B}\ = \frac{1}{\mu_0}\,(1\,-\,(\chi_m\,+\,1)^{-1})\,\mathbf{B}[/tex]

    Extended explanation

    Bound charge and current:

    Electric susceptibility converts [itex]\mathbf{E}[/itex], which acts on the total charge, to [itex]\mathbf{P}[/itex], which acts only on bound charge (charge which can move only locally within a material).

    Magnetic susceptibility converts [itex]\mathbf{H}[/itex], which acts on free current, to [itex]\mathbf{M}[/itex], which acts only on bound current (current in local loops within a material, such as of an electron "orbiting" a nucleus).

    Relative permittivity [itex]\mathbf{\varepsilon_r}[/itex] and relative permeability [itex]\mathbf{\mu_r}[/itex]:

    [tex]\mathbf{\varepsilon_r}\ =\ \mathbf{\chi_e}\ +\ 1[/tex]

    [tex]\mathbf{\mu_r}\ =\ \mathbf{\chi_m}\ -\ 1[/tex]

    [tex]\mathbf{D}\ =\ \varepsilon_0\,\mathbf{E}\ +\ \mathbf{P}\ =\ \varepsilon_0\,(1\,+\,\mathbf{\chi_e})\,\mathbf{E}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E}[/tex]

    [tex]\mathbf{B}\ =\ \mu_0\,(\mathbf{H}\ +\ \mathbf{M})\ =\ \mu_0\,(1\,+\,\mathbf{\chi_m})\,\mathbf{H}\ =\ \mathbf{\mu_r}\,\mathbf{H}[/tex]

    Note that the magnetic equations analogous to [itex]\mathbf{P}\ = \mathbf{\chi_e}\,\varepsilon_0\,\mathbf{E}[/itex] and [itex]\mathbf{D}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E}[/itex] are [itex]\mathbf{M}\ = \frac{1}{\mu_0}\,(1\,-\,(\mathbf{\chi_m}\,+\,1)^{-1})\,\mathbf{B}[/itex] and [itex]\mathbf{H}\ =\ \mathbf{\mu_r}^{-1}\,\mathbf{B}[/itex]

    In other words, the magnetic analogy of relative permittivity is the inverse of relative permeability, and the magnetic analogy of electric susceptibility is the inverse of a part of magnetic susceptibility.


    Permittivity: [itex]\mathbf{\varepsilon}\ =\ \varepsilon_0\,\mathbf{\varepsilon_r}[/itex]

    Permeability: [itex]\mathbf{\mu}\ =\ \mu_0\,\mathbf{\mu_r}[/itex]

    Units:

    Relative permittivity and relative permeability, like susceptibility, are dimensionless (they have no units).

    Permittivity is measured in units of farad per metre ([itex]F.m^{-1}[/itex]).

    Permeability is measured in units of henry per metre ([itex]H.m^{-1}[/itex]) or tesla.metre per amp or newton per amp squared.

    cgs (emu) values:

    Some books which give values of susceptibility use cgs (emu) units for electromagnetism.

    Although susceptibility has no units, there is still a dimensionless difference between cgs and SI values, a constant, [itex]4\pi[/itex]. To convert cgs values to SI, divide by [itex]4\pi[/itex] for electric susceptibility, and multiply by [itex]4\pi[/itex] for magnetic susceptibility.

    Tensor nature of susceptibility:

    For crystals and other non-isotropic material, susceptibility depends on the direction, and changes the direction, and therefore is represented by a tensor.

    For isotropic material, susceptibility is the same in every direction, and [itex]\mathbf{P}[/itex] (or [itex]\mathbf{M}[/itex]) is in the same direction as [itex]\mathbf{E}[/itex] (or [itex]\mathbf{H}[/itex]):

    [tex]\mathbf{P}\ = \varepsilon_0\,\chi_e\,\mathbf{E}[/tex]

    where [itex]\chi_e[/itex] is a multiple of the unit tensor, and therefore is effectively a scalar:


    [tex]P^i\ =\ \varepsilon_0\,\chi_e\,E^i[/tex]​

    Ordinary susceptibility is a tensor (a linear operator whose components form a 3x3 matrix) which converts one vector field to another:

    [tex]P^i\ =\ \varepsilon_0\,\chi_{e\ j}^{\ i}\,E^j[/tex]

    Second-order susceptibility is a tensor (a linear operator whose components form a 3x3x3 "three-dimensional matrix") which converts two copies of one vector field to another:

    [tex]P^i\ =\ \varepsilon_0\,\chi_{e\ \ jk}^{(2)\,i}\,E^j\,E^k[/tex]

    It is used in non-linear optics.

    Susceptibility, being a tensor, is always linear in each of its components. The adjective "non-linear" refers to the presence of two (or more) copies of [itex]\bold{E}[/itex].

    More generally, one can have:

    [tex]P^i\ =\ \varepsilon_0\,\sum_{n\ =\ 1}^{\infty}\chi_{e\ \ \ j_1\cdots j_n}^{(n)\,i}\,E^{j_1}\cdots E^{j_n}[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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