Relative permittivity

In physics, in particular in electrostatics, the relative permittivity (also known by the now obsolescent term dielectric constant) is an intrinsic property of a non-conducting material, a dielectric. It is usually denoted by ε_r. For simple dielectrics, using SI units, ε_r is related to the proportionality constant between the electric field E and the electric displacement D, namely D = ε_r ε₀ E, where ε₀ is the electric constant. The relative permittivity describes the ease by which a dielectric medium may be polarized. The capacitance of a capacitor is proportional to ε_r.

When two electric charges are placed inside a dielectric, the electrostatic force between the charges is changed by a factor 1/ε_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 ε_r ≥ 1. Only classical vacuum has ε_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).

It should be noted that the use of a constant as the relative permittivity of a substance is an approximation, even in the case of a quantum vacuum. A more complete representation recognizes that all media exhibit departures from this approximation, in particular, a dependence on field strength, a dependence upon the rate of variation of the field in both time and space, and a dependence upon the direction of the field. In many materials these dependencies are slight; in others, like ferroelectrics, they are pronounced.

Definition by capacitance of parallel-plate capacitor

An common alternative definition invokes parallel-plate capacitors. In order to make the connection with the definition above, we recall that the electric field above a charged plate of infinite size is independent of the distance from the plate. (This will be shown later in a separate section.) If σ is the charge density on the plate (charge per surface, in SI units C/m²), the strength E of the field E is given by

E = σ/(2 ε₀ε_r),

where ε₀ is the electric constant. In Gaussian units one may take ε₀ = 1. When the charge density σ 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 ε_r, is double that of one plate

E = σ/(ε₀ε_r).

If the plates have surface area A, they carry a total charge Q = σ A (positive on one plate, negative on the other),

Q = ε₀ε_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 ε_r:

C = ε_r C₀ with C₀ ≡ ε₀ A / d.

Clearly, C₀ is the capacity with the reference medium of classical vacuum between the plates, and in principle one may define

$\epsilon _{r}\equiv {\frac {C}{C_{0}}}.$

In practice, the measurement is compared with some real medium that approximates the ideal reference medium of classical vacuum. For example, one could use an ideal gas as the comparison medium and measure the permittivity as the pressure of this gas was reduced. By fitting a theoretical expression relating the permittivity to the gas pressure, one can extrapolate to zero pressure to approximate the comparison to the ideal classical vacuum.

This comparison definition of relative permittivity is common, and reads in words: ε_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 ε_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.

This comparative approach to determining relative permittivity is a macroscopic measurement of dielectrics, an averaged property that can be deduced in principle from the underlying electrical behavior of the material on the atomic scale.

Generalizations

The relative permittivity of a dielectric is a function of temperature and, when the dielectric is a gas, also of pressure. (The pressure dependence of solid and liquid dielectrics is usually very weak.)

For low-symmetry dielectrics (solids and liquid crystals) it may happen that the constant is a second rank tensor, which is represented with respect to a Cartesian coordinate system by a 3 × 3 matrix

${\boldsymbol {\epsilon }}_{r}={\begin{pmatrix}\epsilon _{11}&\epsilon _{12}&\epsilon _{13}\\\epsilon _{21}&\epsilon _{22}&\epsilon _{23}\\\epsilon _{31}&\epsilon _{32}&\epsilon _{33}\\\end{pmatrix}}.$

For frequency-dependent (time-dependent) electric fields the relative permittivity is in general a function of the angular frequency ω. The function ε_r(ω) may vary strongly as function of ω, especially for ω-values in the visible or UV part of the electromagnetic spectrum. It may also happen that a dielectric absorbs energy of an electromagnetic wave that hits it (dielectric loss). This absorption is described by extending the permittivity into the complex plane; the real part of ε_r(ω) has the meaning given above, while its imaginary part describes the energy absorption.

Electric field above infinite plate

To compute E we let the charged 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 corresponding negative components.

Use that an infinitesimal surface element in cylinder coordinates times the surface charge density σ gives an infinitesimal charge in the plate,

$\mathrm {d} Q=\sigma \;r\;\mathrm {d} r\mathrm {d} \varphi \,,$

where σ is assumed constant over the plate and for convenience sake 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

$\mathrm {d} E_{z}={\frac {R_{z}\mathrm {d} Q}{4\pi \epsilon _{0}\epsilon _{r}R^{3}}}={\frac {\sigma zr\mathrm {d} r\mathrm {d} \varphi }{4\pi \epsilon _{0}\epsilon _{r}(r^{2}+z^{2})^{3/2}}}.$

Integration over φ gives a factor 2π and

$E_{z}={\frac {z\sigma }{2\epsilon _{0}\epsilon _{r}}}\int _{0}^{\infty }{\frac {r}{(r^{2}+z^{2})^{3/2}}}\;\mathrm {d} r={\frac {\sigma }{2\epsilon _{0}\epsilon _{r}}}.$

Often one defines the electric displacement by D ≡ ε₀ε_rE. Cleary D_z depends only on the charge density and not on the permittivity,

$D_{z}={\frac {1}{2}}\sigma .$

The fact that E and D do not depend on the distance of the field point to the plate may be compared with the gravitational force, −mg, being independent of height close to the surface of the Earth.