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5.3 Analytic representations of electrodynamic quantities: complex permittivity and conductivity
[electrodynamics top]
Section G.3 in the appendices discusses the fact that, if the time dependence of our voltages are sinusoidal, the
relations between the voltage/current, their integrals, and their derivatives take on a special form. We take advantage
of this by using the analytic representation of the sinusoidal signals, in which case we exchange the real description
of a physical quantity with its analytic representation, which is easier to work with but involves complex
numbers. For hydraulic circuit analysis, this replaces hydraulic resistance and compliance with the
complex hydraulic impedance, which obeys a simple HagenPoiseuilletype relation ( = ). This
highlights that the relative response of a resistor (which impedes flow or current) and a capacitor (which
stores volume or charge) differs by a factor of jω, which means that the flow stored by compliance is
delayed by π∕2 as compared to that through a microchannel, and the ratio of the currents between the
two is proportional to ω. This analysis is mathematically possible because the frequency spectra of
real functions are symmetric around the origin, and was physically possible because the governing
equations for circuits are linear and homogeneous. For the equations of electrodynamics, a similar
approach is possible—this approach makes analysis of system responses to sinusoidal fields enormously
simpler.
Just as we might have for a circuit, we can replace a real electric field = _{0}cosωt with its analytic
representation = _{0}expjωt and replace the real equations of electrodynamics with equivalent analytic
representations. We can furthermore drop the expjωt and write the result strictly in terms of the phasors. For
example, we can (See Exercise 5.8) replace = ε with
 (5.51) 
where is the analytic representation of the electrical displacement. For electrodynamic systems, we
combine conductivity and permittivity into a complex parameter or . In so doing, the constitutive
relations for electric displacement ( = ε) and ohmic current density (= σ) can be combined to
form
 (5.52) 
where the complex permittivity isdefined as
 (5.53) 
Similarly, we can replace = σ with
 (5.54) 
where the complex conductivity isgiven by:
 (5.55) 
5.3.1 Complex description of dielectric loss
For determination of electric charge transport and solution of electrodynamic equations, the complex permittivity
from Equation 5.53 correctly describes both displacement and ohmic current. For describing the conversion of
electric field to heat, though, the conductivity σ describes on one of the loss mechanisms. The energy of an applied
electric field can be directly absorbed by a medium, a process called dielectric loss. Thus conductivity leads to
charge transport and, in turn, heat generation, while dielectric loss leads to heat generation directly. The dielectric
loss for a mode i is proportional to χ_{e,i}(0)ωτ_{i}. We write the applied field E = E_{0}cosωt using its analytic
representation = E_{0}expjωt, and group the polarization response and dielectric loss terms into a complex
permittivity = ε′+ε′′. Here ε′ denotes the real part of the permittivity, i.e., the measureof the polarization
response to the electric field:
 (5.56) 
and the imaginary component ε′′ denotes the(imaginary) dissipation of electrical (polarization) energy into internal
energy of the molecules of the medium:
 (5.57) 
From these relations, we see that ε′ is a measure of how much the medium polarizes in response to an electric field,
while ε′′ is the product of the relative medium response times ωτ_{i}. The energy dissipation is low at high frequency,
because the medium has no time to polarize and no energy is transferred to the medium from the field. The energy
dissipation is low at low frequency, because, though the medium polarizes, the total energy stored per cycle is low.
However, at frequencies near 1∕τ_{i}, the medium is polarized quite strongly (one half of the polarization at
steadystate) and the number of cycles is high. Thus significant energy is pumped into the medium for each
cycle, and there are many cycles per second. The total energy conversion to heat is thus proportional to
σ∕ω+ε′′.
The real and imaginary components of the complexpermittivity are related by the KramersKrĂ¶nig relations,
discussed in Appendix G:
 (5.58) 
and
 (5.59) 
While these relations are exact, they do not immediately lend themselves to an intuitive interpretation. For the
sigmoidal functions that describe the permittivity, the following relation holds, which is exact at ω = 1∕τ_{i} and
approximate for frequencies near ω = 1∕τ_{i}:
 (5.60) 
If plotted on a semilog axis, the dissipative response is approximately equal to the derivative of the reactive
response. Thus energy dissipation coincides with those frequencies where the permittivity is decreasing with ω.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
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