# Permittivity

Physical size
Surname dielectric conductivity or permittivity
Formula symbol ${\ displaystyle \ varepsilon}$
Size and
unit system
unit dimension
SI F · m -1 = A · s · V -1 · m -1 M -1 · L -3 · T 4 · I 2
Gauss ( cgs ) - 1
esE ( cgs ) - 1
emE ( cgs ) c −2 L −2 · T 2

The permittivity ε (from the Latin permittere : allow, leave, allow), also called dielectric conductivity or dielectric function , indicates in electrodynamics and also in electrostatics the polarization ability of a material through electrical fields .

A permittivity is also assigned to the vacuum , since electric fields can also arise in the vacuum or electromagnetic fields can spread. It is a natural constant , namely the electric field constant . The permittivity of a substance is then given as a multiple of the permittivity of the vacuum: ${\ displaystyle \ varepsilon _ {0}}$

${\ displaystyle \ varepsilon = \ varepsilon _ {0} \ varepsilon _ {\ mathrm {r}}}$

The factor here is the substance-dependent relative permittivity. However, it depends not only on the type of substance, but also on the frequency of the active fields, among other things. ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

## Explanation using the example of insulating materials

In a dielectric, the orientation of stationary electrically charged dipoles leads to polarization effects. Such materials can conduct the electrical flow by a factor (relative permittivity) better than empty space.${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

Permittivity is a material property of electrically insulating , polar or non-polar substances, which are also called dielectrics . The property takes effect when the substance interacts with an electric field, for example when it is in a capacitor .

In a capacitor filled with material, the charge carriers of the insulation material are oriented towards the vector of the electric field and generate a polarization field that counteracts the external field and weakens it. Assuming a given electrical excitation field , also called electrical flux density, this phenomenon of field weakening can be described by assigning a factor to the electrical field constant (permittivity of the vacuum) to the insulating material . In a vacuum, the reference material for an insulating material is the relative permittivity${\ displaystyle {\ vec {D}}}$${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$${\ displaystyle \ varepsilon _ {0}}$${\ displaystyle \ varepsilon _ {\ mathrm {r}} = 1}$

From the external electrical excitation the electrical field then results with the permittivity : ${\ displaystyle \ varepsilon = \ varepsilon _ {\ mathrm {r}} \, \ varepsilon _ {0}}$${\ displaystyle {\ vec {E}}}$

${\ displaystyle {\ vec {E}} = {\ frac {\ vec {D}} {\ varepsilon}} = {\ frac {\ vec {D}} {\ varepsilon _ {\ mathrm {r}} \, \ varepsilon _ {0}}}}$

With constant electrical excitation and increasing values ​​of , the electrical field strength decreases. In this way, the field-weakening effect is recorded with the same electrical excitation, i.e. H. for a given electrical flux density or a given electrical charge. ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

Under the influence of a fixed voltage  U applied to the capacitor plates and the electric field (plate spacing  d ), the electrical excitation results with the permittivity as a proportionality factor : ${\ displaystyle E = {\ tfrac {U} {d}}}$

${\ displaystyle \ Leftrightarrow {\ vec {D}} = \ varepsilon {\ vec {E}} = \ varepsilon _ {\ mathrm {r}} \ varepsilon _ {0} {\ vec {E}}}$

The electrical susceptibility is linked to the relative permittivity via ${\ displaystyle \ chi _ {e}}$

${\ displaystyle \ varepsilon _ {\ mathrm {r}} = {\ frac {\ varepsilon} {\ varepsilon _ {0}}} = 1+ \ chi _ {e}}$

The susceptibility is a measure of the density of the charge carriers bound in the insulation material, based on the density of free charge carriers.

According to the Poisson equation of electrostatics, the permittivity can also be viewed as a proportionality factor between the space charge density and the second partial derivative of the potential field: ${\ displaystyle \ rho}$ ${\ displaystyle \ Phi}$

${\ displaystyle \ varepsilon = - {\ frac {\ rho (\ mathbf {r})} {\ Delta \ Phi (\ mathbf {r})}}}$

## Permittivity of the vacuum

The permittivity of the vacuum is a natural constant . In a vacuum there is the following relationship between the magnetic field constant , the permittivity of the vacuum and the vacuum speed of light : ${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle \ mu _ {0}}$${\ displaystyle \ varepsilon _ {0}}$ ${\ displaystyle c_ {0}}$

${\ displaystyle \ varepsilon _ {0} = {\ frac {1} {\ mu _ {0} c_ {0} ^ {2}}} = 8 {,} 854 \, 187 \, 812 \, 8 \, (13) \ cdot 10 ^ {- 12} ~ {\ frac {\ mathrm {As}} {\ mathrm {Vm}}}}$

The unit of permittivity can be expressed as:

${\ displaystyle \ mathrm {F \, m ^ {- 1}} = \ mathrm {A ^ {2} \, s ^ ​​{4} \, kg ^ {- 1} \, m ^ {- 3}} = \ mathrm {As \, V ^ {- 1} \, m ^ {- 1}} = \ mathrm {C \, V ^ {- 1} m ^ {- 1}} = \ mathrm {J \, V ^ {-2} m ^ {- 1}}}$

Since the electrical polarizability of air is low, the permittivity of the air ( ε r  ≈ 1.00059) can often be approximated with sufficient accuracy . This is particularly the case with radar and radio technology . ${\ displaystyle \ varepsilon _ {0}}$

### Numerical value and unit

In addition to the Coulomb's law , the Ampere's law and Faraday law of induction , the relationship between μ 0 , and c another link electromagnetic and mechanical units, which fall in the choice of the electromagnetic unit system is taken into account. ${\ displaystyle \ varepsilon _ {0}}$

Depending on the system of units used, the representation of the permittivity changes analogously to the representation of . ${\ displaystyle \ varepsilon = \ varepsilon _ {0} \ varepsilon _ {\ mathrm {r}}}$${\ displaystyle \ varepsilon _ {0}}$

The relationships in the SI are given above. In systems of units that explicitly trace the electromagnetic quantities back to mechanical basic quantities, namely the different variants of the CGS system of units , the number is chosen as the quantity of the dimension : ${\ displaystyle \ varepsilon _ {0}}$

${\ displaystyle \ varepsilon _ {0}: = 1}$( Heaviside-Lorentz system of units ),
${\ displaystyle \ varepsilon _ {0}: = {\ frac {1} {4 \ pi}}}$(electrostatic, electromagnetic or Gaussian system of units ).

## Relative permittivity

The relative permittivity of a medium (designation according to the DKE-IEV 121-12-13 standard), also called the permittivity or dielectric constant , is the dimensionless ratio of its permittivity to the permittivity of the vacuum: ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon _ {0}}$

${\ displaystyle \ varepsilon _ {\ mathrm {r}} = {\ frac {\ varepsilon} {\ varepsilon _ {0}}}}$

For gaseous, liquid and solid matter it is . However, in other states of matter, e.g. B. in plasma , also values . ${\ displaystyle \ varepsilon _ {\ mathrm {r}}> 1}$${\ displaystyle \ varepsilon _ {\ mathrm {r}} <1}$

The relative permittivity is a measure of the field-weakening effects of the dielectric polarization of the medium. In the English-language literature and in semiconductor technology , the relative permittivity is also referred to as ( kappa ) or - as in the case of high-k dielectrics or low-k dielectrics  - with  k . ${\ displaystyle \ kappa}$

As a synonym for (relative) permittivity, the previous designation (relative) dielectric constant is still in use. The designation as a constant is inappropriate because it is generally a function of several parameters, in particular frequency and temperature; it also depends on the magnetic and external electric field. ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

A scalar quantity is only for isotropic media . In this simplest case, it indicates the factor by which the voltage across a capacitor drops if, with the same geometry, a vacuum assumed between the capacitor electrodes is replaced by a dielectric, non-conductive material. This can be understood in the experiment if a volume of air around the capacitor electrodes z. B. is replaced by a dielectric liquid. For a plate capacitor, it is sufficient to push a dielectric object between the electrodes. ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$

### Directional dependence

In general, the relative permittivity is a second order tensor . This reflects their directional dependence, which results from the crystalline (or otherwise ordered) structure of matter, e.g. B. for birefringent materials that u. a. when retardation plates are used. The tensor property of permittivity is the basis of crystal optics .

In addition to the “natural” directional dependency, the properties can also experience a similar directional dependency through external influences such as a magnetic field (see magneto-optics ) or pressure .

### Frequency dependence

The permittivity of water depends on the frequency:
the real part describes the capacity or refractive index, the imaginary part describes the energy absorption.

The frequency dependence ( dispersion ) of the permittivity in matter can be modeled quite well using the Lorentz oscillator and is e.g. B. very pronounced in water, cf. Illustration.

Like the electrical permittivity, the refractive index of a material also depends on the frequency , since according to Maxwell's equations it has the following relation to the permittivity: ${\ displaystyle n}$${\ displaystyle f}$

${\ displaystyle n ^ {2} (f) = \ varepsilon _ {\ mathrm {r}} (f) \ cdot \ mu _ {\ mathrm {r}} (f)}$

With

Here and μ are meant for the relevant optical frequency (order of magnitude 10 15  Hz). The optical dispersion is an expression that also at the frequencies of visible light is not a constant number. ${\ displaystyle \ varepsilon}$${\ displaystyle \ varepsilon}$

In tables , the numerical value is usually given at low frequencies (order of magnitude 50 Hz to 100 kHz) at which molecular dipoles can follow the external field almost instantaneously.

#### Complex-valued relative permittivity

Course of the complex-valued relative permittivity over a wide frequency range, split into real (red) and imaginary parts (blue) with symbolic representation of the various causes such as relaxation and, at higher frequencies, atomic and electronic resonances

Just as with constant fields, polarization fields also form in dielectrics with alternating fields , but they lag behind the applied external field size by a certain phase angle . This means that the orientation of the charge carriers in the dielectric lags behind the polarization of the alternating field in the phase.

Therefore, the relative permittivity is generally complex valued :

${\ displaystyle \ varepsilon _ {\ mathrm {r}} = \ varepsilon _ {\ mathrm {r}} '+ \ mathrm {i} \ varepsilon _ {\ mathrm {r}}' '}$

or

${\ displaystyle \ varepsilon _ {\ mathrm {r}} = \ varepsilon _ {1} + \ mathrm {i} \ varepsilon _ {2}}$

The contributions of various mechanisms in the material (e.g. band transitions ) can be specified in the real and imaginary parts and their frequency dependence can be added - a more detailed representation can be found under electrical susceptibility .

The lagging effect becomes stronger with increasing frequency . By quickly and repeatedly repolarizing insulating materials, alternating fields of high frequency convert electromagnetic field energy into thermal energy . This heat loss is called dielectric loss and is described by the imaginary part or the complex-valued relative permittivity. ${\ displaystyle \ varepsilon _ {\ mathrm {r}} ''}$${\ displaystyle \ varepsilon _ {2}}$

A widespread application that takes advantage of the dielectric heating phenomenon is in the microwave oven .

In dielectric heating, the loss is power density , based on the volume of material

${\ displaystyle p = {\ frac {P _ {\ mathrm {verl}}} {V}} = \ omega \ cdot \ varepsilon _ {\ mathrm {r}} '' \ cdot \ varepsilon _ {0} \ cdot E ^ {2}}$

with the angular frequency . See also dielectric dissipation factor . ${\ displaystyle \ omega}$

The power loss associated with dielectric heating, when integrated over the heating period, corresponds exactly to the internal energy that was supplied to the material volume with electromagnetic waves, as described in thermodynamics .

At even higher frequencies, with which charge carriers in the band model of a crystal can be excited, energy is also absorbed ( dielectric absorption ).

### Field strength dependence

In the case of high field strengths , the relationship between electric field and flux density becomes non-linear . Either one understands the permittivity as field strength dependent or one introduces besides further Taylor coefficients etc. which describe the field strength dependence of : ${\ displaystyle \ left (\ varepsilon = f (E) \ right)}$${\ displaystyle \ varepsilon ^ {(1)}: = \ varepsilon}$ ${\ displaystyle \ varepsilon ^ {(2)}, \ varepsilon ^ {(3)}}$${\ displaystyle {\ vec {D}}}$

${\ displaystyle {\ vec {D}} = {\ frac {\ vec {E}} {\ | {\ vec {E}} \ |}} \ left (\ varepsilon ^ {(1)} \ cdot \ | {\ vec {E}} \ | + \ varepsilon ^ {(2)} \ cdot \ | {\ vec {E}} \ | ^ {2} + \ varepsilon ^ {(3)} \ cdot \ | {\ vec {E}} \ | ^ {3} + \ cdots \ right)}$

### Temperature dependence

The temperature-dependent, for example, the complex-valued relative permittivity of water, the real part of which assumes a value of around 80 at a frequency of 1 GHz and a temperature of 20 ° C and around 52 at 95 ° C. The decrease in permittivity with increasing temperature is related to the increasing degree of disorder of the charge carriers with an increase in internal energy. From a molecular point of view, the polarizability decreases due to the increasing intrinsic movement of the charge carriers with higher internal energy; From a macroscopic point of view, the relative permittivity decreases with a rise in temperature.

### Values ​​for selected materials

Relative permittivity of some substances
(unless otherwise stated: at 18 ° C and 50 Hz)
medium ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$ medium ${\ displaystyle \ varepsilon _ {\ mathrm {r}}}$
vacuum exactly 1 air 1,00059
Acrylonitrile Butadiene Styrene (ABS) (30 ° C) 4.3 Aluminum oxide (clay) 9
Ammonia (0 ° C) 1.007 Barium titanate 10 3 … 10 4
benzene 2.28 Dry earth 3.9
Damp earth 29 Germanium 16.6
Glass 6… 8 Glycerin 42.5
rubber 2.5 ... 3 Wood ( dry ) 2… 3.5
Potassium chloride 4.94 Methanol 32.6
petroleum 2 Polyethylene (PE) (90 ° C) 2.4
Polypropylene (PP) (90 ° C) 2.1 porcelain 2… 6
Propanol 18.3 paraffin 2.2
paper 1… 4 Polytetrafluoroethylene
( PTFE or Teflon)
2
FR2 , FR4 4.3 ... 5.4 Polystyrene foam
(Styropor ® BASF )
1.03
Tantalum pentoxide 27 Water (20 ° C, 0… 3 GHz) 80
Water (visible area) 1.77 Water (0 ° C, 0… 1 GHz) 88
Ice (0 to −50 ° C, low frequency) ≈ 90 ... 150 Ice (over 100 kHz) 3.2

Tabulated, comprehensive overviews of frequency- and temperature-dependent, complex relative permittivities of many materials can be found in Industrial Microwave Heating and especially in Dielectric Materials and Applications .

## Relationship with absorption and reflection

Using the Kramers-Kronig relation , the dispersive relationship between the complex permittivity and the optical parameters refractive index and absorption coefficient  k can be shown:

${\ displaystyle (n + \ mathrm {i} k) ^ {2} = (\ varepsilon _ {\ mathrm {r}} '+ \ mathrm {i} \ varepsilon _ {\ mathrm {r}}' ') \ mu _ {\ mathrm {r}}}$

In the case of non-magnetic materials ( ) it follows after a coefficient comparison : ${\ displaystyle \ mu _ {\ mathrm {r}} \ approx 1}$

${\ displaystyle \ Rightarrow \ varepsilon _ {\ mathrm {r}} '= n ^ {2} -k ^ {2}}$
${\ displaystyle \ varepsilon _ {\ mathrm {r}} '' = 2nk}$

For the calculation of theoretical spectra of reflection and absorption , which can be compared with measured spectra and adapted, the components of the complex refractive index must be determined directly from the real and imaginary part of the permittivity:

${\ displaystyle n ^ {2} = {\ frac {1} {2}} \ cdot \ left ({\ sqrt {\ varepsilon _ {\ mathrm {r}} '^ {2} + \ varepsilon _ {\ mathrm {r}} ^ {\ prime \ prime 2}}} + \ varepsilon _ {\ mathrm {r}} '\ right)}$
${\ displaystyle k ^ {2} = {\ frac {1} {2}} \ cdot \ left ({\ sqrt {\ varepsilon _ {\ mathrm {r}} '^ {2} + \ varepsilon _ {\ mathrm {r}} ^ {\ prime \ prime 2}}} - \ varepsilon _ {\ mathrm {r}} '\ right)}$

Also can u. a. the reflectance  R can be calculated for a ray coming from the vacuum (or air) which is reflected perpendicularly at an interface to a medium with a refractive index : ${\ displaystyle N = n + \ mathrm {i} k}$

${\ displaystyle R = {\ frac {(n-1) ^ {2} + k ^ {2}} {(n + 1) ^ {2} + k ^ {2}}}}$

## literature

• Richard P. Feynman , Robert B. Leighton , Matthew Sands : The Feynman Lectures on Physics. Volume 2: Mainly Electromagnetism and Matter. 6th printing. Addison-Wesley, Reading MA et al. a. 1977, ISBN 0-201-02117-X .
• Heinrich Frohne : Introduction to Electrical Engineering. Volume 2: Heinrich Frohne, Erwin Ueckert: Electric and magnetic fields. (= Teubner study scripts. Vol. 2: Electrical engineering. ). 4th revised edition. Teubner, Stuttgart 1983, ISBN 3-519-30002-8 .
• Arthur von Hippel : Dielectrics and Waves . Wiley et al. a., New York NY u. a. 1954 (2nd edition. Artech House, Boston MA et al. 1995, ISBN 0-89006-803-8 ).
• Arthur von Hippel (Ed.): Dielectric Materials and Applications . Technology Press, Boston MA et al. a. 1954, ISBN 0-89006-805-4 (2nd edition. Artech House, Boston MA et al. 1995).
• AC Metaxas: Foundations of Electroheat. A unified approach . John Wiley and Sons, Chichester et al. a. 1996, ISBN 0-471-95644-9 .
• AC Metaxas, RJ Meredith: Industrial Microwave Heating (=  IEE Power Engineering Series . Vol. 4). Peter Peregrinus, London 1983, ISBN 0-906048-89-3 .
• Károly Simonyi : Theoretical electrical engineering . 10th edition. Barth Verlagsgesellschaft, Leipzig a. a. 1993, ISBN 3-335-00375-6 .

## Individual evidence

1. DKE-IEV German online edition of the IEV. (aspx page, enter "relative permittivity" in the search field). Retrieved May 23, 2020 .
2. Martin Chaplin: Water and Microwaves. Water Structure and Science, accessed July 9, 2018.
3. ^ AC Metaxas, RJ Meredith: Industrial Microwave Heating (=  IEE Power Engineering Series . Vol. 4). Peter Peregrinus, London 1983, ISBN 0-906048-89-3 .
4. Arthur von Hippel (Ed.): Dielectric Materials and Applications . Technology Press, Boston MA et al. a. 1954 (2nd edition. Artech House, Boston MA et al. 1995, ISBN 0-89006-805-4 ).