Editing Dielectric (section)

===Debye relaxation===
'''Debye relaxation''' is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity ''ε'' of a medium as a function of the field's [[angular frequency]] ''ω'':

<math display="block">\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{1 + i\omega\tau},</math>

where ''ε<sub>∞</sub>'' is the permittivity at the high frequency limit, {{nowrap|Δ''ε'' {{=}} ''ε<sub>s</sub>'' − ''ε<sub>∞</sub>''}} where ''ε<sub>s</sub>'' is the static, low frequency permittivity, and ''τ'' is the characteristic [[relaxation time]] of the medium. Separating into the real part <math>\varepsilon'</math> and the imaginary part <math>\varepsilon''</math> of the complex dielectric permittivity yields:<ref>{{cite book|title=Dielectric Phenomena in Solids|last=Kao|first=Kwan Chi|publisher=Elsevier Academic Press|year=2004|isbn=978-0-12-396561-5|location=London|pages=92–93}}</ref>

<math display="block">\begin{align}
   \varepsilon' &= \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1 + \omega^2\tau^2} \\[3pt]
  \varepsilon'' &= \frac{(\varepsilon_s - \varepsilon_\infty)\omega\tau}{1+\omega^2\tau^2}
\end{align}</math>

Note that the above equation for <math>\hat{\varepsilon}(\omega)</math> is sometimes written with <math>1 - i\omega\tau</math> in the denominator due to an ongoing sign convention ambiguity whereby many sources represent the time dependence of the complex electric field with <math>\exp(-i\omega t)</math> whereas others use <math>\exp(+i\omega t)</math>. In the former convention, the functions <math>\varepsilon'</math> and <math>\varepsilon''</math> representing real and imaginary parts are given by <math>\hat{\varepsilon}(\omega)=\varepsilon'+ i \varepsilon''</math> whereas in the latter convention <math>\hat{\varepsilon}(\omega)=\varepsilon'- i \varepsilon''</math>. The above equation uses the latter convention.<ref>{{cite book|title=Theory of Electric Polarisation|last=Böttcher|first=C.J.F.|publisher=Elsevier Publishing Companys|year=1952|location=London|pages=231–232, 348–349}}</ref>

The dielectric loss is also represented by the loss tangent:

<math display="block">\tan(\delta) = \frac{\varepsilon''}{\varepsilon'} = \frac{\left(\varepsilon_s - \varepsilon_\infty\right)\omega\tau}{\varepsilon_s + \varepsilon_\infty \omega^2 \tau^2}</math>

This relaxation model was introduced by and named after the physicist [[Peter Debye]] (1913).<ref>Debye, P. (1913), Ver. Deut. Phys. Gesell. 15, 777; reprinted 1954 in collected papers of Peter J.W. Debye. Interscience, New York</ref> It is characteristic for dynamic polarisation with only one relaxation time.