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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. 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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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] [Induced-Charge Effects] [DEP] [Solution Chemistry]

5.1 Electrostatics in matter [electrodynamics top]

VIDEO: electrostatic fundamentals with view toward electroosmosis.

Electrostatics involves the effects of stationary charges or static electric fields on a test charge, and amounts to various statements and restatements of Coulomb’s law. The electrostatic limit applies when charges are stationary and the current is zero.

We study in charges in matter, usually in an aqueous solution (we also call this an electrolyte solution) or in a metal conductor. In matter, it becomes unwieldy to keep track of all of the electrostatic interactions. To simplify things, we distinguish betweenfree charge andbound charge, and keep detailed track of only the free charge. Free charge implies a charge that is mobile over distances large as compared to atomic length scales. Free charge typically comes from electrons (in a metal) or ions (in an aqueous solution). We treat free charges specifically in the sections to follow. Bound charge implies charges of equal magnitude but opposite signs that are held in close proximity and are free to move only atomic distances (roughly 1 Å or less). Examples of bound charge include the positive charge of an atomic nucleus and the negative charge of its associated electron cloud, the uneven partial charges in a heteronuclear covalent bond, or the bound ions in a solid crystal. Rather than calculate all of the details of bound charge with detailed electrostatic equations, we replace the detailed effects of bound charge through the continuum electrical permittivity.

Since microscale flows are often driven by voltages specified by power supplies, we begin our description with the electrical potential and electric field.

5.1.1 Electrical potential and electric field

We consider a point charge q andthe force it feels owing to the presence of other charges. We define the electrical potential ϕ (also called thevoltage) at a point in space such that the electrostatic potential energy that a point charge has with respect to a reference position in given by . We define the electric field microfluidics textbook nanofluidics textbook Brian Kirby Cornell suchthat the force on a point charge q is given by microfluidics textbook nanofluidics textbook Brian Kirby Cornell = qmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

5.1.2 Coulomb’s Law, Gauss’s Law for electricity in a material, curl of electric field

Given a static source charge q embedded in a linear, instantaneously-responsive, isotropic material of electricpermittivity ε, Coulomb’s law describes the electric field induced in the material by this charge:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell is the distance vector from the point charge to the location in question, microfluidics textbook nanofluidics textbook Brian Kirby Cornell  is the unit vector in this direction (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby CornellΔr), r is the magnitude of this vector, and ε [CV meter] is the electrical permittivity of the medium. We can similarly write the electrical potential caused by a point charge in matter:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.3)

Because these relations are for a charge embedded in matter, they take into account both the source charge and the response of all of the bound charge in the material. In treating the medium as linear and isotropic, we are neglecting nonlinear effects seen at high fields, as well as anisotropy observed, for example, in some crystals. Equation 5.2 implicitly treats the material as responding instantaneously. We address finite response times later in the chapter when we describe the frequency dependence of the permittivity.

The flux of the electric field caused by an ensemble of point charges can be integrated over a surface, deriving Gauss’s law forelectricity:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Here, the integral is performed over a closed surface S, microfluidics textbook nanofluidics textbook Brian Kirby Cornell is a unit outward normal along this surface, dA is a differential area along this surface, ε is the electrical permittivity of the material, and q is the sum of the charges [C] inside the surface. By applying the divergence theorem, this integral relation can be put in differential form:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where ρE is the volumetric net free charge density in the material [Cm3]. We refer to ρE simply asthe charge density in solution.

By calculating the line integral of the electric field around a closed contour, we can also show that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.6)

where ds is a differential line element. This can be converted to differential form with Stokes’ theorem,leading to

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The irrotational property of the electric field is consistent with the earlier definition of the electric field (Equation 5.1) in terms of the gradient of a scalar potential, since the curl of the gradient of a scalar is zero by definition.

Gauss’s law can also be written as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where microfluidics textbook nanofluidics textbook Brian Kirby Cornell is the electric displacement or electric flux density [Cm2], defined as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

While microfluidics textbook nanofluidics textbook Brian Kirby Cornell and microfluidics textbook nanofluidics textbook Brian Kirby Cornell can be converted back and forth quite easily, their analytical purposes are different. Theelectric displacement (through ∇⋅microfluidics textbook nanofluidics textbook Brian Kirby Cornell = ρE) tells us what the effects of a source charge are, while the electric field (through microfluidics textbook nanofluidics textbook Brian Kirby Cornell = qmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell) tells us what the effects on a test charge are. In vacuum, the two properties have the same value. In matter, (especially high-εr materials like water), the two values are different.

5.1.3 Polarization of matter and electric permittivity

A source charge in vacuum induces an electric field, as defined by Coulomb’s law and thepermittivity of free space ε0. The electrical permittivity of free space ε0 is a fundamental constant, and is given byε0 = 8.85×10-12 CVm.

When a source charge is embedded in matter, we divide the effects of the source charge into two parts: (a) thepolarization of the medium, which we treat as a continuum, and (b) the residual electric field that exists despite the polarization of the matter. Thus, we can think of all electric fields in matter (especially in electrolyte solutions, since water’s electric permittivity is large) as being residual electric fields, the small leftover field after most of the effects of the source charge are canceled by the polarization of the matter. Gauss’s law for electricity in matter lumps the two effects of the source charge (polarization and residual electric field) into one term, the electric displacement.

The electric polarization microfluidics textbook nanofluidics textbook Brian Kirby Cornell inmatter is the dipole moment per unit volume, and measures the degree to which bound charge is polarized. The polarizability of a medium is described by theelectric susceptibility χe, which is a function of the atom-scale structure of the material as well as the temperature:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.10)

The electrical permittivity [CVm] of a material reflects the sum total of the polarization response of the matter plus the residual electric field, and is given by ε = ε0(1+χe). Electrical permittivities for several materials are given in Table 5.1. The relative permittivity or dielectric constant, definedas
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.11)

is used to denote the ratio of the total response (both polarization and residual electric field) of a particular medium to the electric field alone.

The terminology here can be confusing. The electric susceptibility tells us how much a medium polarizes in response to an electric field. Free space has nothing to polarize, and its electric susceptibility is zero, whereas water has a large dipole, and its electric susceptibility is large. The electric permittivity tells us how much electric field is caused by a source charge, and is useful when we want to relate fields to charge with Gauss’s law. As compared to vacuum, the dielectric constant tells us how much smaller the electric fields are in a medium if the same source charge is used.

A common misconception is that the dielectric constant indicates how much more a medium polarizes than a vacuum—this is incorrect, because a vacuum cannot polarize. Water’s dielectric constant is 80 not because water polarizes 80 times more than space; rather, the dielectric constant is 80 because when water polarizes, the electric field that is left over is 1/80th as big as it would have been if the water remained unpolarized. The electric susceptibility, similarly, does not indicate that water polarizes 79 times more than space—it indicates that the field caused by the polarized water is 79 times bigger than the residual electric field.





Material χ e εr



vacuum 0 1



dry air 5×10-4 1.0005



dodecane 1 2



glass 5 6



silicon 11 12



isopropanol 17 18



ethanol 23 24



methanol 32 33



acetonitrile 36 37



water 79 80




Table 5.1: Electric susceptibilities and electric permittivities for several materials.

For example, consider a (positively charged) sodium ion dissolved in water. As seen in Figure 5.1, the sodium ion causes the polar water molecules to orient themselves in response to the electric field, and would cause an attractive force on a negatively charged ion (for example, a chloride ion). The presence of the water molecules tends to decrease the electric field caused by the ions, because the polarized water molecules generate their own electric field in the opposite direction. The net effect is that the test charge (the chloride ion) feels an attractive force, but a much smaller one than would be felt if the ions were in a vacuum. As we have written it, Coulomb’s law treats the free charges (sodium, chloride) on an individual, atomistic level, but subsumes the effects of bound charges (water OH bonds) into a continuum with a continuum property ε.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.1: Polarization of matter (in this case, water) in response to charged ions. Bold lines: the electric field lines that the sodium and chloride ions would have generated if they were in a vacuum. The polar water molecules orient in response to the ions, with the oxygen (carrying a partial negative charge) toward the sodium atom and the hydrogens (carrying partial positive charges) toward the chloride atom. The orientation of the polar water molecules causes its own electric field (dashed lines), which cancels out most (microfluidics textbook nanofluidics textbook Brian Kirby Cornell) of the electric field that would exist if the ions were in a vacuum. The resulting net electric field is well approximated by the field given by Coulomb’s law for charges in matter, Equation 5.2. Ions and molecules are not drawn to scale. The extent of the polarization is exaggerated in this figure—in fact, the linear material assumption in Equation 5.9 describes matter response to electric fields only if the induced polarization is a small perturbation of the state of the material.


Charge density of an electrolyte solution

We are concerned with the local netfree charge density of the fluids we are studying, which are typically aqueous solutions. Ignoring the solvent, which is typically neutral, the net free charge density ρE can be related to species concentrations by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.12)

where ci is the molar concentration of species i, zi is thevalence (charge normalized by the elementary charge) of species i, and F isFaraday’s constant, equal to F = eNA = 96485 Cmol. The free charge density thus corresponds (in liquid systems) to the density of charge of mobile ions. The bound charge has been accounted for by the continuum concept of the electrical permittivity, as discussed earlier. In the absence of external forces, the potential energy of any system of ions is minimized if the net charge is distributed evenly. Thus most media are electroneutral, i.e., the net charge density at all points within domains is zero–any charge is located only at surfaces. This is true for perfect conductors. In the fluid systems we are considering, which have finite conductivity and are also affected by diffusive forces stemming from the thermal energy of the solvent molecules, the fluid has a nonzero net charge density within a finite region near walls, specified by the Debye length λD, defined in Chapter 9.

5.1.4 Material, frequency, and electric field dependence of electrical permittivity

Materials experience anelectric polarization in an electric field, as described by Equation 5.10. The electric polarization is a measure of the degree to which the bound charge in a medium polarizes, with positive components being pulled in the direction of the electric field and negative components being pulled in the direction opposite to theelectric field. Multiple polarization mechanisms contribute to this phenomenon, and these mechanisms each have a different characteristic time. In the presence of an electric field, the electron cloud of an atom is displaced with respect to the nucleus, leading to electronic polarization, while atomic bonds stretch, leading to atomic polarization. At 100 Vcm, this deflection is roughly 1 billionth of an atomic radius. These phenomena are rapid, occurring over a characteristic time approximately equal to 1×10-15–1×10-18 s. Molecules with permanent dipoles orient rotationally(owing to the torque, see Section 5.1.10), leading to a net orientational polarization. This is the contribution that makes water’s permittivity so large as compared to other liquids. While atomic polarization requires motion on subatomic length scales (much smaller than 1 Å), dipole orientation requires motion on atomic length scales (1 Å or higher), which takes longer. The effective rotational relaxation time for water at room temperature, for example, is about 8 ps. Orientational polarization only occurs if the frequency of the electric field is below 1×1012 Hz. Figure 5.2 shows these modes of polarization schematically.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.2: Polarization modes.


Debye relaxation model for frequency dependence of permittivity Equation 5.9 assumes that the response of the medium to an applied field is instantaneous. However, the polarization requires a finite time. We find it most convenient, then, to decompose the applied field into sinusoidal components and treat the permittivity as a frequency-dependent property. For applied fields whose frequency is much higher than the characteristic frequency of the polarization process, the polarization process is largely ineffective, and thus that process does not contribute to the permittivity at that frequency. For those applied fields whose frequency is much lower than the characteristic frequency of the polarization process, the polarization process contributes to the permittivity at that frequency. We thus think of ε(ω) in terms of the sum of different physical processes. Each process has a characteristictime τ that signifies the speed of the process—for example, τ for atomic polarization is approximately 1.0×10-15 s, and τ for orientational polarization is approximately 1.2×10-11 s. In the Debye model of permittivity, the contribution to the electric susceptibility from each polarization mechanism i takes the form
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.13)

where ω is the frequency of the applied field, χe,i(0) is the contribution of mechanism i to the polarization if the applied field is steady, and τi is the characteristic time of polarization mechanism i. This expression is analogous to the expression for the frequency dependence of the power transmitted through a capacitor in an RC circuit. The total electric susceptibility for a material is thus given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.14)

and the permittivity is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.15)

For water at room temperature, for example, χe,i for atomic polarization is approximately 5, while χe,i for orientational polarization is approximately 72, so an approximation for the permittivity of water at room temperature is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.16)

The Debye model assumes that the response of the medium is made up of noninteracting dipoles with a single response time for orientation. This model is a good approximate model, but typically must be replaced with a model that is more sophisticated or has more free parameters, if experimental data are to be well predicted. For orientational polarization, the Debye model does not account for interactions between molecules and the attendant spread of response times; for atomic and electronic polarization, the Debye model is inaccurate because the dipoles are created by the field rather than oriented by the field. Thus the predictions of Equation 5.16 should be considered qualitative. A graph of the dependence on the electric permittivity and the dielectric loss is shown in Figure 5.3.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.3: Frequency dependence of electric permittivity and dielectric loss.


Electric field dependence of permittivity The use of a permittivity as described above implies that the medium is linear, meaning that double the electric field leads to double the polarization. This is correct for electronic and atomic polarization of water. This approximation is correct for orientation of water only if the orientational energy of the inherent dipole moment of each molecule in the field is small as comparedto kBT , meaning that the statistical orientation of the molecules is minor. Here, kB is theBoltzmann constant (1.38×10-23JK). For water, when the applied electric field is on the order of 1×109 Vm, the orientational energy pE kBT is of the same order as the thermal energy of the system, and the water is largely aligned. Once the alignment is significant, the linear model for the response of the material is no longer accurate. An approximate description of the permittivity of water as a function of the electric field uses the Brillouin functionto approximate the statistical vector average of the dipole moment of the water molecules as a function of applied field:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.17)

where p is the magnitude of thedipole moment of an individual water molecule (2.95 D), microfluidics textbook nanofluidics textbook Brian Kirby Cornell denotes the statistical average dipole moment of an ensemble of water molecules, E is the magnitude of the applied electric field, and B denotes the Brillouin function for dipole orientation with two possible states, i.e., B(x) = 2coth2x-cothx. From Equation 5.17, the ensemble-averaged dipole moment of water molecules varies linearly at low fields (since B(x) (x) for small x), but approaches a constant (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = p) as the electric field E approaches infinity. Since the ensemble-averaged dipole moment cannot increase beyond p, the permittivity is proportional to 1∕E as E approaches infinity. The approximate description of the permittivity is thus given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.18)

where the 6 includes atomic and electronic polarization, and the 72 corresponds to the orientational polarization. This relation is approximate, and ignores other nonlinear effects such as electrostriction.

5.1.5 Poisson and Laplace equations

The Poisson and Laplace equations are simplified forms of Gauss’s law that apply if no time-varying magnetic fields are present. From above, Gauss’s law for electricity is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.19)

Considering irrotational electric fields and writing theelectric field as the gradient of the electrical potential (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = -∇ϕ), we get thePoisson equation:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.20)

written here for spatially varying ε. For uniform ε, this can be rewritten as:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The Laplace equationis derived from the Poisson equation if ρE is assumed to be zero:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

which is the case for electrolyte solutions far away from solid walls.

5.1.6 Classification of material types

We classify materials into several categories based on their response to electric fields. Ideal dielectrics or, equivalently, perfect insulators arematerials that have no free charge–their charge is all bound at the atomic level. These materials cannot conduct current, so their conductivity σ = 0, but they do polarize in response to electric fields and thus they have an electrical permittivity ε. Perfect conductors or ideal conductors havecharges (typically electrons) that are assumed free to move in the material. The electric field and charge density in a perfect conductor are both zero, and thus the conductor is an equipotential and is electroneutral. Regardless of any externally-applied electric field, the charge in a conductor aligns itself at the edges of the conductor to perfectly cancel out this field, and all of the net charge in a conductor is located on the surface of the conductor. Perfect conductors have σ = , and, because they experience no electric field, the permittivity of perfect conductors is undefined. In between these two limits, materials that carry finite current and support a finite electric field are called weak conductors or lossy dielectrics. These materials have finite conductivity σ and finite permittivity ε. In microfluidic systems that employ electric fields, the device substrate is typically an insulator (e.g., glass or polymer) or has an insulating coating (such as a silicon oxide coating on a silicon microdevice), electrodes are made from conducting materials, and the working fluid is typically an aqueous solution that we treat as a lossy dielectric. Thus microsystem boundaries are usually perfect insulators or perfect conductors, while the governing equations are used to study weakly conducting fluids.

5.1.7 Electrostatic boundary conditions

For the electrostatic case (i.e., where all charges are static), Gauss’s lawcan be applied to the boundary between two domains to determine the boundary conditions for the normal and tangential components of the electric field. Given an interface with a charge per unit areagiven by q′′, separating two domains labeled 1 and 2 with permittivities of ε1 and ε2, respectively, the boundary condition for the normal component of the electric field is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.23)

The direction of microfluidics textbook nanofluidics textbook Brian Kirby Cornell points from domain 1 to domain 2, as shown in Figure 5.5. A line integral over a loop at the boundary gives the boundary condition for any tangent component of the electric field:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.24)

Equation 5.24 holds for any unit vector tangent to the surface.

Boundary conditions at microdevice walls and inlets We are generally concernedwith the boundary conditions for the electric field for a weakly conducting fluid at a solid surface, either an insulating surface (such as a polymer or glass) or a conducting surface (such as a metal electrode). Because the fluid channels in a microdevice are usually connected to large reservoirs that are, in turn, connected to electrode wires, we often need to treat the inlets and outlets of a device as boundaries in an analytical or computational treatment of the electrostatic equations.

For an insulating wall that has a surface charge density q′′, Equation 5.23 is simplified because the electric field in the insulating wall (microfluidics textbook nanofluidics textbook Brian Kirby Cornell1) normal to the surface is approximately zero. Thus the normal boundary condition is a relation between the surface charge density and the potential gradient normal to the surface:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.25)

where n is the coordinate normal to the surface, pointing into the weakly conducting fluid.

For a conducting wall such as a metal electrode, the metal surface provides a constant-potential boundarycondition
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.26)

Rigorous treatment of inlets and outlets to microchannels requires specification of the entire reservoir geometry. However, since microdevice reservoirs are typically enormous as compared to the microchannels themselves, we neglect any voltage drop between the electrode and the inlet to the microchannel. Thus, in the electrostatic case,
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.27)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(5.28)

Figure 5.4 depicts these boundary conditions for a sample microdevice.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.4: Upper left: schematic of a glass microdevice with inlet and outlet reservoirs with potentials specified by electrical connections made with platinum wires. A microchannel connects the inlet and outlet, and micropatterned electrodes on the bottom of the microchannel are independently addressed. Lower right: schematic of the microchannel fluid domain and the electrostatic boundary conditions in the limit without current or ion motion. The glass surface is treated as an insulating surface with a surface charge density, the electrode is treated as a conductor and therefore an isopotential, and the inlet and outlet are approximated as being equal to the potential applied to the reservoir.



microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.5: The boundary between two domains microfluidics textbook nanofluidics textbook Brian Kirby Cornell points from domain 1 to domain 2. The tangential boundary condition applies for any microfluidics textbook nanofluidics textbook Brian Kirby Cornell  vector.


5.1.8 Solution of electrostatic equations

The electrostatic equations are typically solved numerically. For systems that can be approximated as having charge only at interfaces, the Laplace equation applies and thus conformal mapping is applicable. For microchannel systems that can be approximated as having polygonal cross-sections that vary slowly along their axis, Schwarz-Christoffel transforms enable rapid analytical solution (Chapter G).

5.1.9 Maxwell stress tensor

Fundamentally, if magnetic fields are steady and if the volume is not moving quickly, the electrical forces on a particle are simply the sum of all Coulomb interactions between charges and fields. Since Maxwell’s equations relate charges and fields, the force on a control volume can be put into a form that is a function of the fields only. This is particularly useful when we know the electric fields along the surface of a controlled volume better than we know the distribution of charge inside the volume. To calculate the force on a volume, we introduce the Maxwell stress tensormicrofluidics textbook nanofluidics textbook Brian Kirby Cornell:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

which, in the absence of magnetic fields, is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

With these definitions, the force on a volumecan be written in terms of the Maxwell stress tensor:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The Maxwell stress tensor is a dyadic tensor; see Section C.2.5 in the appendix. The relation between the force and the Maxwell stress tensor is useful in Chapter 17 for determining the force on a microparticle as a function of the electric fields applied to it.

5.1.10 Effects of electrostatic fields on multipoles

The multipolar theory is used to describe electricity and magnetism for both mathematical and physical reasons. Mathematically, multipolarsolutions (see Appendix F) arise naturally from use of separation of variables to analytically solve the Laplace equation. Physically, some of these solutions correspond to physical objects that play a central role in electrodynamic systems, such as point charges and magnetic dipoles. Here we summarize relations for forces and torques on monopoles (point charges) and dipoles. In all cases, the force is written in terms of the applied electric field at the location of the multipole. By applied electric field, we imply the field that would have been at that point if the multipole were absent.

Electrical force on a monopole(i.e., point charge)

The electrical force on a point charge is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

This is shown schematically in Figure 5.6.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.6: Force on a monopole.


Electrical force on a dipole

Theelectrical force on a dipole in a nonuniform field can be found by summing the forces on its constituent monopoles. This is shown schematically in Figure 5.7. The result can be written in terms of the dipole moment microfluidics textbook nanofluidics textbook Brian Kirby Cornell [C m]:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The dipole moment is measured in units of Coulomb-meters or Debye (where 1 D = 3.33×10-30 C m).


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 5.7: Force on a dipole.


Electrical torque on a dipole

The electrical torque on a dipole in a nonuniform field can be found by summing the forces on its constituent monopoles (Figure 5.7):

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

[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] [Induced-Charge 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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