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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 \chi_e is a measure of the ease of polarisation of a material.
Magnetic susceptibility \chi_m 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 \chi_e and magnetic susceptibility \chi_m are the operators which convert the electric field and the magnetic intensity field, \varepsilon_0\mathbf{E} and \mathbf{H} (not the magnetic field \mathbf{B}), respectively, to the polarisation and magnetisation fields \mathbf{P} and \mathbf{M}:
\mathbf{P}\ = \chi_e\,\varepsilon_0\,\mathbf{E}
\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}
Extended explanation
Bound charge and current:
Electric susceptibility converts \mathbf{E}, which acts on the total charge, to \mathbf{P}, which acts only on bound charge (charge which can move only locally within a material).
Magnetic susceptibility converts \mathbf{H}, which acts on free current, to \mathbf{M}, which acts only on bound current (current in local loops within a material, such as of an electron "orbiting" a nucleus).
Relative permittivity \mathbf{\varepsilon_r} and relative permeability \mathbf{\mu_r}:
\mathbf{\varepsilon_r}\ =\ \mathbf{\chi_e}\ +\ 1
\mathbf{\mu_r}\ =\ \mathbf{\chi_m}\ -\ 1
\mathbf{D}\ =\ \varepsilon_0\,\mathbf{E}\ +\ \mathbf{P}\ =\ \varepsilon_0\,(1\,+\,\mathbf{\chi_e})\,\mathbf{E}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E}
\mathbf{B}\ =\ \mu_0\,(\mathbf{H}\ +\ \mathbf{M})\ =\ \mu_0\,(1\,+\,\mathbf{\chi_m})\,\mathbf{H}\ =\ \mathbf{\mu_r}\,\mathbf{H}
Note that the magnetic equations analogous to \mathbf{P}\ = \mathbf{\chi_e}\,\varepsilon_0\,\mathbf{E} and \mathbf{D}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E} are \mathbf{M}\ = \frac{1}{\mu_0}\,(1\,-\,(\mathbf{\chi_m}\,+\,1)^{-1})\,\mathbf{B} and \mathbf{H}\ =\ \mathbf{\mu_r}^{-1}\,\mathbf{B}
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: \mathbf{\varepsilon}\ =\ \varepsilon_0\,\mathbf{\varepsilon_r}
Permeability: \mathbf{\mu}\ =\ \mu_0\,\mathbf{\mu_r}
Units:
Relative permittivity and relative permeability, like susceptibility, are dimensionless (they have no units).
Permittivity is measured in units of farad per metre (F.m^{-1}).
Permeability is measured in units of henry per metre (H.m^{-1}) 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, 4\pi. To convert cgs values to SI, divide by 4\pi for electric susceptibility, and multiply by 4\pi 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.
Ordinary susceptibility is a tensor (a linear operator whose components form a 3x3 matrix) which converts one vector field to another:
P^i\ =\ \varepsilon_0\,\chi_{e\ j}^{\ i}\,E^j
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:
P^i\ =\ \varepsilon_0\,\chi_{e\ \ jk}^{(2)\,i}\,E^j\,E^k
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 \bold{E}.
More generally, one can have:
P^i\ =\ \varepsilon_0\,\sum_{n\ =\ 1}^{\infty}\chi_{e\ \ \ j_1\cdots j_n}^{(n)\,i}\,E^{j_1}\cdots E^{j_n}
* 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!
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 \chi_e is a measure of the ease of polarisation of a material.
Magnetic susceptibility \chi_m 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 \chi_e and magnetic susceptibility \chi_m are the operators which convert the electric field and the magnetic intensity field, \varepsilon_0\mathbf{E} and \mathbf{H} (not the magnetic field \mathbf{B}), respectively, to the polarisation and magnetisation fields \mathbf{P} and \mathbf{M}:
\mathbf{P}\ = \chi_e\,\varepsilon_0\,\mathbf{E}
\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}
Extended explanation
Bound charge and current:
Electric susceptibility converts \mathbf{E}, which acts on the total charge, to \mathbf{P}, which acts only on bound charge (charge which can move only locally within a material).
Magnetic susceptibility converts \mathbf{H}, which acts on free current, to \mathbf{M}, which acts only on bound current (current in local loops within a material, such as of an electron "orbiting" a nucleus).
Relative permittivity \mathbf{\varepsilon_r} and relative permeability \mathbf{\mu_r}:
\mathbf{\varepsilon_r}\ =\ \mathbf{\chi_e}\ +\ 1
\mathbf{\mu_r}\ =\ \mathbf{\chi_m}\ -\ 1
\mathbf{D}\ =\ \varepsilon_0\,\mathbf{E}\ +\ \mathbf{P}\ =\ \varepsilon_0\,(1\,+\,\mathbf{\chi_e})\,\mathbf{E}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E}
\mathbf{B}\ =\ \mu_0\,(\mathbf{H}\ +\ \mathbf{M})\ =\ \mu_0\,(1\,+\,\mathbf{\chi_m})\,\mathbf{H}\ =\ \mathbf{\mu_r}\,\mathbf{H}
Note that the magnetic equations analogous to \mathbf{P}\ = \mathbf{\chi_e}\,\varepsilon_0\,\mathbf{E} and \mathbf{D}\ =\ \mathbf{\varepsilon_r}\,\mathbf{E} are \mathbf{M}\ = \frac{1}{\mu_0}\,(1\,-\,(\mathbf{\chi_m}\,+\,1)^{-1})\,\mathbf{B} and \mathbf{H}\ =\ \mathbf{\mu_r}^{-1}\,\mathbf{B}
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: \mathbf{\varepsilon}\ =\ \varepsilon_0\,\mathbf{\varepsilon_r}
Permeability: \mathbf{\mu}\ =\ \mu_0\,\mathbf{\mu_r}
Units:
Relative permittivity and relative permeability, like susceptibility, are dimensionless (they have no units).
Permittivity is measured in units of farad per metre (F.m^{-1}).
Permeability is measured in units of henry per metre (H.m^{-1}) 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, 4\pi. To convert cgs values to SI, divide by 4\pi for electric susceptibility, and multiply by 4\pi 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 \mathbf{P} (or \mathbf{M}) is in the same direction as \mathbf{E} (or \mathbf{H}):
\mathbf{P}\ = \varepsilon_0\,\chi_e\,\mathbf{E}
where \chi_e is a multiple of the unit tensor, and therefore is effectively a scalar:
P^i\ =\ \varepsilon_0\,\chi_e\,E^i
\mathbf{P}\ = \varepsilon_0\,\chi_e\,\mathbf{E}
where \chi_e is a multiple of the unit tensor, and therefore is effectively a scalar:
P^i\ =\ \varepsilon_0\,\chi_e\,E^i
Ordinary susceptibility is a tensor (a linear operator whose components form a 3x3 matrix) which converts one vector field to another:
P^i\ =\ \varepsilon_0\,\chi_{e\ j}^{\ i}\,E^j
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:
P^i\ =\ \varepsilon_0\,\chi_{e\ \ jk}^{(2)\,i}\,E^j\,E^k
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 \bold{E}.
More generally, one can have:
P^i\ =\ \varepsilon_0\,\sum_{n\ =\ 1}^{\infty}\chi_{e\ \ \ j_1\cdots j_n}^{(n)\,i}\,E^{j_1}\cdots E^{j_n}
* 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!
