What is susceptibility

1. Jul 24, 2014

Greg Bernhardt

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.

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:

$$P^i\ =\ \varepsilon_0\,\chi_e\,E^i$$​

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:

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

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