Editing Permittivity (section)

=== Lossy media ===
In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

<math display="block">J_\text{tot}\ =\ J_\mathrm{c} + J_\mathrm{d} = \sigma\ E\ +\ i\ \omega\ \varepsilon'\ E = i\ \omega\ \hat{\varepsilon}\ E\ </math>

where
* {{mvar|σ}} is the [[electrical conductivity|conductivity]] of the medium;
* <math>\ \varepsilon'\ =\ \varepsilon_0\ \varepsilon_\mathsf{r}\ </math> is the real part of the permittivity.
* <math>\ \hat{\varepsilon}\ =\ \varepsilon' - i\ \varepsilon''\ </math> is the complex permittivity

Note that this is using the electrical engineering convention of the [[Mathematical descriptions of opacity#Complex conjugate ambiguity|complex conjugate ambiguity]]; the physics/chemistry convention involves the complex conjugate of these equations.

The size of the [[displacement current]] is dependent on the [[frequency]] {{mvar|ω}} of the applied field {{mvar|E}}; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as:<ref>
{{cite book
 |first=John S. |last=Seybold
 |year=2005
 |title=Introduction to RF Propagation
 |publisher=John Wiley & Sons
 |page=22, eq.&nbsp;(2.6)
 |isbn=9780471743682
 |url=https://books.google.com/books?id=4LtmjGNwOPIC
}}
</ref><ref>
{{cite book
 |first=Kenneth L. |last=Kaiser
 |title=Electromagnetic Shielding
 |publisher=CRC Press
 |year=2005
 |pages=1–28, eqs.&nbsp;(1.80) and (1.81)
 |isbn=9780849363726
 |url=https://books.google.com/books?id=bDuOAQDk38gC
}}
</ref>

<math display="block">\ \hat{\varepsilon}\ =\ \varepsilon' \left(\ 1\ -\ i\ \frac{\sigma}{\ \omega \varepsilon'\ } \ \right)\ =\ \varepsilon'\ -\ i\ \frac{\ \sigma\ }{\ \omega\ } </math>

In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:

* First are the [[Dielectric relaxation|relaxation]] effects associated with permanent and induced [[Dipole|molecular dipoles]]. At low frequencies the field changes slowly enough to allow dipoles to reach [[wikt:equilibrium|equilibrium]] before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the [[viscosity]] of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called [[dielectric relaxation]] and for ideal dipoles is described by classic [[Debye relaxation]].
* Second are the [[resonance|resonance effects]], which arise from the rotations or vibrations of atoms, [[ion]]s, or [[electron]]s. These processes are observed in the neighborhood of their characteristic [[Absorption (electromagnetic radiation)|absorption frequencies]].

The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called ''soakage'' or ''battery action''. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1–2% of the original voltage. However, it can be as much as 15–25% in the case of [[electrolytic capacitor]]s or [[supercapacitor]]s.