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[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
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[Solution Chemistry]
In electrical circuits, we consider idealized elements and simple relations that relate voltage and current. Below, we
list circuit components, discuss complex representations of real properties, discuss notational concepts such as
impedance and phasors, and summarize circuit relations. This discussion is a description of electrostatics and
electrodynamics for discretized elements with welldefined electrical properties.
5.4.1 Components and properties
Key idealized circuit components include voltage sources, wires, resistors, capacitors, and inductors. Voltage
sources are objects that specify a voltage V [V] at a given point. Ideal wires link together regions of space that
have the same voltage V or electrical potential. Ideal resistors(Figure 5.12) are objects with a finite
resistance to the motion of charge, defined by theresistance R [Ω] and a voltagecurrent relation given
by
 (5.61) 
Idealcapacitors (Figure 5.13) are physical systems with a finite capacity to carry charge, denoted by
acapacitance C [F] and a voltagecurrent relation given by I = CΔV . An ideal capacitor consists of two conductors
separated by a dielectric material–the two conductors each are at a specific voltage and the polarization of the
dielectric causes a net charge on the surface of the conductors.
Idealinductors (Figure 5.14) havean inductance L [H] and a voltagecurrent relation given by ΔV = LI.
We construct electrical circuits by wiring together voltage sources with these resistors, capacitors, and
inductors.
5.4.2 Electrical impedance
The impedanceof a circuit element is a complex quantity that extendsOhm’s law (V = IR) to AC circuits. If the
voltages and currents are sinusoidal and written as follows (note Re means the real part of):
 (5.62) 
 (5.63) 
then for circuit elements we define an impedance , which describes the voltagecurrent relationship through the
equation = or _{0} = _{0}. Each circuit element has a complex impedance corresponding to its circuit
properties:
 resistors:
 capacitors:
 inductors:
5.4.3 Circuit relations
Ohm’s law and the relations for impedance describe the current through an element or the voltage drop across that
element. For a circuit that is composed of a network of theseelements, Kirchoff’s current law links these network
elements using theconservation of current relation (Figure 5.15):
where I in this case is defined positive into the node. Circuit networks can be solved as systems of algebraic
equations constructed fromOhm’s law and circuit element impedance relations.
5.4.4 Series and parallel component rules
The results of Kirchoff’s law for parallel and series circuits are shown below.
Series circuit rules
The resistance of two resistors in series is equal to the sum of the resistances:
the reciprocal of the capacitance of two capacitors in series is equal to the sum of the reciprocals of the
capacitances:
the inductance of two inductors in series is equal to the sum of the inductances:
and the impedance of two impedances in series is equal to the sum of the impedances:
Series component relations are depicted in Figure 5.16.
Parallel circuit rules
The reciprocal of the resistance of two resistors in parallel is equal to the sum of the reciprocals of the
resistances:
the capacitance of two capacitors in parallel is equal to the sum of the capacitances:
the reciprocal of the inductance of two inductors in parallel is equal to the sum of the reciprocals of the
inductances:
and the reciprocal of the impedance of two impedances in parallel is equal to the sum of the reciprocals of the
impedances:
Parallel component relations are depicted in Figure 5.17.
[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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