Dielectric

q

{\displaystyle q}

) should have units of force*
*Electric field (

E

{\displaystyle E}

) should have units of force per area (the same as stress)*
*Electric constant (

ϵ
0

{\displaystyle \epsilon\_0}

) should have units of force per area (the same as bulk modulus)*
*Magnetic field (

B

{\displaystyle B}

) should have units of force per velocity per time^2*
*Magnetic constant (

μ
0

{\displaystyle \mu\_0}

) should have units of 1/(force per time^2)*
See this for more details

A **dielectric** (or **dielectric material**) is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

The study of dielectric properties concerns storage and dissipation of electric and magnetic energy in materials. Dielectrics are important for explaining various phenomena in electronics, optics, solid-state physics, and cell biophysics.

## Terminology

Although the term *insulator* implies low electrical conduction, *dielectric* typically means materials with a high polarizability. The latter is expressed by a number called the relative permittivity. The term insulator is generally used to indicate electrical obstruction while the term dielectric is used to indicate the energy storing capacity of the material (by means of polarization). A common example of a dielectric is the electrically insulating material between the metallic plates of a capacitor. The polarization of the dielectric by the applied electric field increases the capacitor's surface charge for the given electric field strength.

A *perfect dielectric* is a material with zero electrical conductivity (cf. perfect conductor infinite electrical conductivity), thus exhibiting only a displacement current; therefore it stores and returns electrical energy as if it were an ideal capacitor.

## Electric susceptibility

Main articles: Electric susceptibility, Permittivity
The electric susceptibility *?e* of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

It is defined as the constant of proportionality (which may be a tensor) relating an electric field **E** to the induced dielectric polarization density **P** such that

P

=

ε
0

χ
e

E

,

{\displaystyle \mathbf{P} = \varepsilon\_0 \chi\_e \mathbf{E},}

where *e*0 is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity *er* by

χ
e

=

ε
r

−

{\displaystyle \chi\_e\ = \varepsilon\_r - 1.}

So in the case of a vacuum,

χ
e

=

{\displaystyle \chi\_e\ = 0.}

The electric displacement **D** is related to the polarization density **P** by

D

=

ε
0

E

P

=

ε
0

(

1

χ
e

)

E

=

ε
0

ε
r

E

.

{\displaystyle \mathbf{D} \ = \
 \varepsilon\_0 \mathbf{E} + \mathbf{P} \ = \
 \varepsilon\_0 \left(1 + \chi\_e\right) \mathbf{E} \ = \
 \varepsilon\_0 \varepsilon\_r \mathbf{E}.
}

### Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field. The more general formulation as a function of time is

P

(
t
# )

ε
0

∫

−
∞

t

χ
e

(

t
−

t
′

)

E

(

t
′

)

d

t
′

.

{\displaystyle \mathbf{P}(t) = \varepsilon\_0 \int\_{-\infty}^t \chi\_e\left(t - t'\right) \mathbf{E}\left(t'\right)\, dt'.}

That is, the polarization is a convolution of the electric field at previous times with time-dependent susceptibility given by *?e*(?*t*). The upper limit of this integral can be extended to infinity as well if one defines *?e*(?*t*) = 0 for ?*t* < 0. An instantaneous response corresponds to Dirac delta function susceptibility *?e*(?*t*) = *?ed*(?*t*).

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,

P

(
ω
# )

ε
0

χ
e

(
ω
)

E

(
ω
)
.

{\displaystyle \mathbf{P}(\omega) = \varepsilon\_0 \chi\_e(\omega) \mathbf{E}(\omega).}

The susceptibility (or equivalently the permittivity) is frequency dependent. The change of susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e., *?e*(?*t*) = 0 for ?*t* < 0), a consequence of causality, imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility *?e*(*?*).

## Dielectric polarization

In the classical approach to the dielectric model, a material is made up of atoms. Each atom consists of a cloud of negative charge (electrons) bound to and surrounding a positive point charge at its centre. In the presence of an electric field the charge cloud is distorted, as shown in the top right of the figure.

This can be reduced to a simple dipole using the superposition principle. A dipole is characterized by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled *M*. It is the relationship between the electric field and the dipole moment that gives rise to the behavior of the dielectric. (Note that the dipole moment points in the same direction as the electric field in the figure. This isn't always the case, and is a major simplification, but is true for many materials.)

When the electric field is removed the atom returns to its original state. The time required to do so is the so-called relaxation time; an exponential decay.

This is the essence of the model in physics. The behavior of the dielectric now depends on the situation. The more complicated the situation, the richer the model must be to accurately describe the behavior. Important questions are:

* Is the electric field constant or does it vary with time? At what rate?
* Does the response depend on the direction of the applied field (isotropy of the material)?
* Is the response the same everywhere (homogeneity of the material)?
* Do any boundaries or interfaces have to be taken into account?
* Is the response linear with respect to the field, or are there nonlinearities?

The relationship between the electric field **E** and the dipole moment **M** gives rise to the behavior of the dielectric, which, for a given material, can be characterized by the function **F** defined by the equation:

M

=

F

(

E

)

{\displaystyle \mathbf{M} = \mathbf{F}(\mathbf{E})}

.
When both the type of electric field and the type of material have been defined, one then chooses the simplest function *F* that correctly predicts the phenomena of interest. Examples of phenomena that can be so modeled include:

* Refractive index
* Group velocity dispersion
* Birefringence
* Self-focusing
* Harmonic generation