Relative permittivity

In physics, in particular in electrostatics, relative permittivity (also known by the now obsolete term dielectric constant) is an intrinsic property, usually denoted by &epsilon;r, of a non-conducting (electrically insulating) material: a dielectric.

When two electric charges are placed inside a dielectric, the electrostatic force between the charges is changed by a factor 1/&epsilon;r. Empirically it is observed that the force between two charges never increases (the force stays the same or decreases) by the presence of any dielectric medium, hence  the relative permittivity  &epsilon;r &ge; 1. Only the vacuum has &epsilon;r = 1 (exact); all dielectrics have values larger than one (but some materials, such as non-dense inert gases, have relative permittivities very close to the vacuum value of unity).

Definition by capacitance of parallel-plate capacitor
An common alternative definition invokes parallel-plate capacitors. In order to make the connection, it must be first pointed out that it follows from Coulomb's law that  the electric field above a charged plate of infinite size is independent of the distance from the plate. (This will be shown in the next section.) If &sigma; is the charge density on the plate (charge per surface, in SI units C/m2), it will be shown that the strength E of the field E is given by
 * E = &sigma;/(2 &epsilon;0&epsilon;r),

where &epsilon;0 is the electric constant. In Gaussian units one may take &epsilon;0 = 1. When the charge density &sigma; is positive the electric field (a vector) E points away from the plate. Of course, plates of infinite size do not exist, but this formula is applicable when the height is much smaller than the dimensions of the plate, so that border effects can be neglected.

In parallel-plate capacitors border effects can usually be ignored and because both plates have the same charge density (but of opposite sign), the electric field inside a capacitor, filled with a dielectric with &epsilon;r, is double that of one plate
 * E = &sigma;/(&epsilon;0&epsilon;r).

If the plates have surface area A, they carry a total charge Q = &sigma; A (positive on one plate, negative on the other),
 * Q = &epsilon;0&epsilon;r A E.

Say the distance between the plates is d, then the voltage difference V between the plates is E / d.

The capacitance C of a capacitor is by definition Q / V, so that we find that the capacitance of a parallel-plate capacitor is linear in the relative permittivity &epsilon;r:
 * C = &epsilon;r C0 with C0 &equiv; &epsilon;0 A / d.

Clearly, C0 is the capacity with vacuum between the plates, and
 * &epsilon;r = C/C0.

This property of capacitors is often used as the definition of relative permittivity: &epsilon;r is equal to the ratio of the capacitance of a capacitor filled with the dielectric to the capacitance of an identical capacitor in a vacuum without the dielectric material.

Because &epsilon;r > 1, the insertion of a dielectric between the plates of a parallel-plate capacitor always increases its capacitance, or ability to store opposite charges on each plate, compared with this ability when the plates are separated by a vacuum.

The relative permittivity is defined as a macroscopic property of dielectrics, without need of specifying the electrical behavior of the material on the atomic scale.

Electric field above infinite plate
Use cylinder coordinates and let the plate be in the x-y plane. It is a priori clear from the symmetry of the problem that the field at a point z on the z-axis has only a z component, the field components parallel to the plate are always compensated by a corresponding negative component. In cylinder coordinates an infinitesimal surface element in the plate has charge

\mathrm{d}Q = \sigma\; r\; \mathrm{d}r\mathrm{d}\phi\,, $$ where the surface density &sigma; is constant over the plate and for convenience we take it positive (hence the field points in the positive z direction). The electric field in the z direction at a point (0,0,z) is by Coulomb's law

\mathrm{d}E_z = \frac{\sigma z r \mathrm{d}r\mathrm{d}\phi}{4\pi \epsilon_0 \epsilon_r (r^2+z^2)^{3/2}}. $$ Integration over &phi; gives a factor 2&pi; and

E_z = \frac{z\sigma}{2 \epsilon_0 \epsilon_r}\int_0^\infty \frac{ r }{(r^2+z^2)^{3/2}}\; \mathrm{d}r =\frac{\sigma}{2 \epsilon_0 \epsilon_r}. $$