Fundamentals of Engineering Electromagnetics - Chapter 2 pdf

Another form of charge separation occurs when avoltage is applied between two conductors, for example the electrodes of a capacitor.Capacitive structures obey the relationship where the

Despite its many useful applications, electrostatic charge is often a nuisance to beavoided For example, sparks of electrostatic origin trigger countless accidental explosionsevery year and lead to loss of life and property Less dramatically, static sparks candamage manufactured products such as electronic circuits, photographic film, and thin-coated materials The transient voltage and current of a single spark event, called an

billion-dollar industry specializing in the prevention or neutralization of ESD-producingelectrostatic charge has of necessity evolved within the semiconductor industry to helpmitigate this problem

Unwanted electrostatic charge can also affect the production of textiles or plastics.Sheets of these materials, called webs, are produced on rollers at high speed Electrostatic

Boston, Massachusetts

microfluidic ‘‘lab on a chip.’’ These microdevices have opened up new vistas of discovery

53

charge can cause webs to cling to rollers and jam production lines Similarly, the sparksthat result from accumulated charge can damage the product itself, either by exposinglight-sensitive surfaces or by puncturing the body of the web.

This chapter presents the fundamentals that one needs in order to understandelectrostatics as both friend and foe We first define the electrostatic regime in the broadcontext of Maxwell’s equations and review several fundamental concepts, includingCoulomb’s law, force-energy relations, triboelectrification, induction charging, particleelectrification, and dielectric breakdown We then examine several applications ofelectrostatics in science and industry and discuss some of the methods used to moderatethe effects of unwanted charge

Like all of electromagnetics, electrostatics is governed by Maxwell’s equations, the elegantmathematical statements that form the basis for all that is covered in this book Trueelectrostatic systems are those in which all time derivatives in Maxwell’s equations areexactly zero and in which forces of magnetic origin are absent This limiting definitionexcludes numerous practical electrostatic-based applications Fortunately, it can berelaxed while still capturing the salient features of the electrostatic domain The

fields and charge magnitudes may vary with time but in which the forces due to the electricfield always dominate over the forces due to the magnetic field At any given moment intime, an electroquasistatic field is identical to the field that would be produced were therelevant charges fixed at their instantaneous values and locations

In order for a system to be electroquasistatic, two conditions must be true: First, anycurrents that flow within the system must be so small that the magnetic fields they producegenerate negligible forces compared to coulombic forces Second, any time variations inthe electric field (or the charges that produce them) must occur so slowly that the effects ofany induced magnetic fields are negligible In this limit, the curl of E approaches zero, andthe cross-coupling between E and H that would otherwise give rise to propagating waves isnegligible Thus one manifestation of the electroquasistatic regime is that the sources ofthe electric field produce no propagating waves

The conditions for satisfying the electroquasistatic limit also can be quantified viadimensional analysis The curl operator r
has the dimensions of a reciprocal distance

L, while each time derivative dt in Maxwell’s equations has the dimensions of a time t.Thus, considering Faraday’s law:

This same dimensional argument can be applied to Ampere’s law:

is much smaller than the propagation wavelength at the frequency of excitation

In the true electrostatic limit, the time derivatives are exactly zero, and Faraday’s law

The curl-free electric field Eq (2.7) can be expressed as the gradient of a scalarpotential :

when energized to a voltage V Applying Gauss’ law to the inner surface of the either

the medium between the electrodes In other, more complex geometries, the solutions toEqs (2.9) and (2.10) take on different forms, as discussed in the next section

When two conductors are connected to a voltage source, one will acquire positive chargeand the other an equal magnitude of negative charge The charge per unit voltage is calledthe capacitance of the electrode system and can be described by the relationship

Figure 2.1 A simple system consisting of two parallel electrodes of area A separated by adistance d

Here Q are the magnitudes of the positive and negative charges, and V is the voltageapplied to the conductors It is easily shown that the capacitance between two parallelplane electrodes of area A and separation d is given approximately by

C ¼"A

where " is the permittivity of the material between the electrodes, and the approximationresults because field enhancements, or ‘‘fringing effects,’’ at the edges of the electrodeshave been ignored Although Eq (2.16) is limited to planar electrodes, it illustrates thefollowing basic form of the formula for capacitance in any geometry:

The source of electrostatic charge lies at the atomic level, where a nucleus having a fixednumber of positive protons is surrounded by a cloud of orbiting electrons The number ofprotons in the nucleus gives the atom its unique identity as an element An individual atom

is fundamentally charge neutral, but not all electrons are tightly bound to the nucleus.Some electrons, particularly those in outer orbitals, are easily removed from individualatoms In conductors such as copper, aluminum, or gold, the outer electrons are weaklybound to the atom and are free to roam about the crystalline matrix that makes up thematerial These free electrons can readily contribute to the flow of electricity In insulatorssuch as plastics, wood, glass, and ceramics, the outer electrons remain bound to individualatoms, and virtually none are free to contribute to the flow of electricity

Electrostatic phenomena become important when an imbalance exists betweenpositive and negative charges in some region of interest Sometimes such an imbalanceoccurs due to the phenomenon of contact electrification [1–8] When dissimilar materialscome into contact and are then separated, one material tends to retain more electrons andbecome negatively charged, while the other gives up electrons and become positivelycharged This contact electrification phenomenon, called triboelectrification, occurs at the

energized electrodes in several different geometries

2.1

points of intimate material contact The amount of charge transferred to any given contactpoint is related to the work function of the materials The process is enhanced by frictionwhich increases the net contact surface area Charge separation occurs on both conductorsand insulators, but in the former case it becomes significant only when at least one of theconductors is electrically isolated and able to retain the separated charge This situation iscommonly encountered, for example, in the handling of conducting powders If neitherconductor is isolated, an electrical pathway will exist between them, and the separatedcharges will flow together and neutralize one another In the case of insulators, however,the separated charges cannot easily flow, and the surfaces of the separated objects remaincharged The widespread use of insulators such as plastics and ceramics in industry andmanufacturing ensures that triboelectrification will occur in numerous situations.The pneumatic transport of insulating particles such as plastic pellets, petrochemicals,fertilizers, and grains are particularly susceptible to tribocharging.

Table 2.1 Field, Potential, and Capacitance Expressions for Various Electrode Geometries

Cylindrical

Vlnðb=aÞln

br



C ¼ 2"hlnðb=aÞ

h  a

The relative propensity of materials to become charged following contact andseparation has traditionally been summarized by the triboelectric series of Table 2.3 (Tribo

is a Greek prefix meaning frictional.) After a contact-and-separation event, the materialthat is listed higher in the series will tend to become positively charged, while the one that

is lower in the series will tend to become negatively charged The vagueness of the phrase

‘‘will tend to’’ in the previous sentence is intentional Despite the seemingly reliable orderimplied by the triboelectric series, the polarities of tribocharged materials often cannot bepredicted reliably, particularly if the materials lie near each other in the series Thisimprecision is evident in the various sources [9–13] cited in Table 2.3 that differ on theexact order of the series Contact charging is an imprecise science that is driven by effects

Table 2.2 Relative Permittivities of Various Materials

Source: Compiled from several sources [9–13].

occurring on an atomic scale The slightest trace of surface impurities or altered surfacestates can cause a material to deviate from the predictions implied by the triboelectricseries Two contact events that seem similar on the macroscopic level can yield entirelydifferent results if they are dissimilar on the microscopic level Thus contact and separation

of like materials can sometimes lead to charging if the contacting surfaces areprobabilistic prediction of polarity during multiple charge separation events Only whentwo materials are located at extremes of the series can their polarities be predicted reliablyfollowing a contact-charging event

The term static electricity invokes an image of charge that cannot flow because it is heldstationary by one or more insulators The ability of charge to be static in fact does depend

on the presence of an insulator to hold it in place What materials can really be consideredinsulators, however, depends on one’s point of view Those who work with electrostaticsknow that the arrival of a cold, dry winter is synonymous with the onset of ‘‘staticseason,’’ because electrostatic-related problems are exacerbated by a lack of humidity.When cold air enters a building and is warmed, its relative humidity declines noticeably.The tendency of hydroscopic surfaces to absorb moisture, thereby increasing their surfaceconductivities, is sharply curtailed, and the decay of triboelectric charges to ground oversurface-conducting pathways is slowed dramatically Regardless of humidity level,however, these conducting pathways always exist to some degree, even under the driest

of conditions Additionally, surface contaminants such as dust, oils, or residues can add tosurface conduction, so that eventually all electrostatic charge finds its way back to ground.Thus, in most situations of practical relevance, no true insulator exists In electrostatics,the definition of an insulator really depends on how long one is willing to wait Statedsuccinctly, if one waits long enough, everything will look like a perfect conductor sooner

or later An important parameter associated with ‘‘static electricity’’ is its relaxation timeconstant—the time it takes for separated charges to recombine by flowing over conductingpathways This relaxation time, be it measured in seconds, hours, or days, must always becompared to time intervals of interest in any given situation

As discussed in the previous section, contact electrification can result in the separation ofcharge between two dissimilar materials Another form of charge separation occurs when avoltage is applied between two conductors, for example the electrodes of a capacitor.Capacitive structures obey the relationship

where the positive and negative charges appear on the surfaces of the opposing electrodes.The electrode which is at the higher potential will carry þQ; the electrode at the lowerpotential will carry Q The mode of charge separation inherent to capacitive structures isknown as inductive charging As Eq (2.18) suggests, the magnitude of the inductivelyseparated charge can be controlled by altering either C or V This feature of induction

charging lies in contrast to triboelectrification, where the degree of charge separation oftendepends more on chance than on mechanisms that can be controlled.

If a conductor charged by induction is subsequently disconnected from its source ofvoltage, the now electrically floating conductor will retain its acquired charge regardless ofits position relative to other conductors This mode of induction charging is used often inindustry to charge atomized droplets of conducting liquids The sequence of diagramsshown in Fig 2.2 illustrates the process The dispensed liquid becomes part the capacitiveelectrode as it emerges from the hollow tube and is charged by induction As the dropletbreaks off, it retains its charge, thereafter becoming a free, charged droplet A droplet of agiven size can be charged only to the maximum Raleigh limit [9,14,15]:

Qmax¼8 ffiffiffiffiffiffiffi

"0

p

the value at which self repulsion of the charge overcomes the surface tension holding thedroplet together, causing the droplet to break up

Nature is fundamentally charge neutral, but when charges are separated by anymechanism, the maximum quantity of charge is limited by the phenomenon of dielectric

by the maximum field magnitude that can be sustained before a field-stressed materialloses its insulating properties.* When a solid is stressed by an electric field, imperfections

Figure 2.2 Charging a conducting liquid droplet by induction As the droplet breaks off (d), itretains the charge induced on it by the opposing electrode

*Breakdown in vacuum invariably occurs over the surfaces of insulating structures used to supportopposing electrodes

or stray impurities can initiate a local discharge, which degrades the composition

of the material The process eventually extends completely through the material, leading

to irreversible breakdown and the formation of a conducting bridge through whichcurrent can flow, often with dramatic results In air and other gases, ever-presentstray electrons (produced randomly, for example, by ionizing cosmic rays) will accelerate

in an electric field, sometimes gaining sufficient energy between collisions to ionizeneutral molecules, thereby liberating more electrons If the field is of sufficientmagnitude, the sequence of ensuing collisions can grow exponentially in a self-sustaining avalanche process Once enough electrons have been liberated from theirmolecules, the gas becomes locally conducting, resulting in a spark discharge Thisphenomenon is familiar to anyone who has walked across a carpet on a dry day and thentouched a doorknob or light switch The human body, having become electrified withexcess charge, induces a strong electric field on the metal object as it is approached,ultimately resulting in the transfer of charge via a rapid, energetic spark The mostdramatic manifestation of this type of discharge is the phenomenon of atmosphericlightning

A good rule of thumb is that air at standard temperature and pressure will break

increases substantially for small air gaps of 50 mm or less because the gap distanceapproaches the mean free path for collisions, and fewer ionizing events take place Hence alarger field is required to cause enough ionization to initiate an avalanche breakdown Thisphenomena, known as the Paschen effect, results in a breakdown-field versus gap-distancecurve such as the one shown in Fig 2.3 [9,10,12,18,19] The Paschen effect is critical to theoperation of micro-electromechanical systems, or MEMS, because fields in excess of

30 kV/cm are required to produce the forces needed to move structural elements madefrom silicon or other materials

Figure 2.3 Paschen breakdown field vs gap spacing for air at 1 atmosphere For large gapspacings, the curve is asymptotic to 3
106V/m

2.9 CORONA DISCHARGE

One of the more common methods for intentionally producing electrostatic chargeinvolves the phenomenon of corona discharge Corona is a partial breakdown that occurswhen two electrodes, one sharp and the other much less so, are energized by a voltagesource In such a configuration, the electric field around the sharp electrode is greatlyenhanced At some critical level of voltage, called the onset voltage, the field near the sharpelectrode exceeds the dielectric breakdown strength of the gas, typically air This localizedbreakdown produces free electrons and positive ions via the avalanche process In theremainder of the electrode space, however, the field is substantially weaker, and noionization takes place Thus the breakdown that occurs near the stressed electrodeprovides a source of ions, but no spark discharge occurs If the stressed electrode ispositive, the positive ions will be repelled from it, providing an abundant source of positiveions If the stressed electrode is negative, the free electrons will be repelled from it but willquickly attach to neutral molecules upon leaving the high field region, thereby formingnegative ions The phenomenon of corona is illustrated graphically in Fig 2.4 for apositive source electrode

For either ion polarity, and in most electrode configurations, the relationshipbetween applied voltage and the resulting corona current follows an equation of the form

geometry, spacing, radii of curvature, and surface roughness, as well as on ion mobility,

in coaxial geometry this relationship can be solved analytically [18] The result is acomplex formula, but for small currents, the equation for cylindrical geometry can beapproximated by

iL¼4"0VðV  VCÞ

temperature and pressure) As the applied voltage V is increased, corona will first occur at

Figure 2.4 Basic mechanism of corona discharge near a highly stressed electrode Positivecorona is shown; a similar situation exists for negative corona

the voltage at which the electric field on the surface of the inner electrode first reaches thevalue given by Peek’s equation [18,20]:

Epeek¼mEbk 1 þ0:0308ffiffiffi

ap

ð2:21Þ

is the inner conductor radius in meters, m is an empirical surface roughness factor, and

inner conductor surface before local breakdown (corona) can occur The equation is alsoapproximately valid for parallel-wire lines For smooth conductors m ¼ 1, and for roughsurfaces m ¼ 0.8

In a coaxial system, the electric field magnitude at the inner radius a is given by

governed by Coulomb’s law, a fundamental principle of physics:

f12 ¼ q1q2

Figure 2.5 Plot of corona onset voltage Vcvs inner conductor radius a for coaxial electrodes with10-cm outer conductor radius

The direction of this force is parallel to a line between the charges All other forcerelationships in electrostatics derive from Coulomb’s law If a collection of chargesproduces a net electric field E, it is easily shown by integration that the collective forceexerted on a solitary charge q by all the other charges becomes just qE This simplerelationship comprises the electric field term in the Lorentz force law of electromagnetics:

In many practical situations in electrostatics, one is interested in the forces on conductorsand insulators upon which charges reside Numerous mathematical methods exist forpredicting such forces, including the force-energy method, the boundary element method,and the Maxwell stress tensor [21–24] Of these three methods, the force-energy method isthe one most easily understood from basic principles and the most practical to use in manysituations The analysis that follows represents an abridged derivation using the force-energy method

We first consider a constant-charge system in which two objects carrying fixed

isolated, the work transferred to the displaced body must increase the energy stored in the

If the electrodes are precharged, then disconnected from their source of voltage, thecharge will thereafter remain constant The stored electrical energy can then be expressed

ddx

Equation (2.29) also describes the force between two insulating surfaces of area A that

It is readily shown [21–24] that applying the energy method to two conductors leftconnected to the energizing voltage V yields a similar force equation:

a system in which voltage, not charge, is constrained, the force it predicts will always beattractive

Equation (2.30) is readily applied to the parallel-electrode structure of Fig 2.7 withthe switch closed The force between the conductors becomes

"Ax

¼ "AV2

This force is inversely proportional to the square of the separation distance x

Many electrostatic processes use the coulomb force to influence the transport of chargedairborne particles Examples include electrostatic paint spraying [10,16], electrostaticFigure 2.7 Parallel electrodes are energized by a voltage source that is subsequently disconnected.Fixed charges Q remain on the electrodes

powder coating, electrostatic crop spraying [25,26], electrostatic drug delivery, andelectrostatic precipitation These processes are described later in this chapter Airborneparticles are sometimes charged by induction, requiring that initial contact be made with aconducting electrode In other processes, particles are charged by ions in the presence of anelectric field.

In this section, we examine the latter process in more detail To a firstapproximation, many airborne particles can be treated as conducting spheres—anassumption that greatly simplifies the equations governing particle charging Theapproximation requires that the particle have a shape free from prominent asymmetriesand also that the intrinsic charging time of the particle, given by the ratio "/ of theparticle’s permittivity to conductivity, be much shorter than other time scales of interest

does not change spatially over the scale of at least several particle radii Further supposethat a uniform, homogeneous source of unipolar ions is produced by the system andcarried toward the particle by the electric field These ions might be produced, for example,

by some form of corona discharge If we assume the ion density to be small enough suchthat space-charge perturbation of the field is negligible, the electric field components in theneighborhood of the particle become:

3 p

points outward

Figure 2.8 Conducting sphere distorts an otherwise uniform electric field The field componentsare given by Eq (2.32) If the source of the field produces ions, the latter will follow the field lines tothe particle surface

perpendicular to the particle surface, where E¼0 Ions will be transported to the surface

of the particle by the field, thereby increasing the magnitude of Q If the ions are positive,only field lines leading into the particle will contribute to its charging Field lines thatoriginate from the surface of the particle cannot carry ions, because no source of ions

becomes larger, the field pattern for Q 6¼ 0 takes the form shown in Fig 2.9 The reduction

in magnitude of the inward-pointing field lines restricts the flow of ions to the particlesurface When Q/4"0r2in Eq (2.32) becomes equal to the factor Eoð1 þ 2R3

p=r3Þat r ¼ Rp,all field lines will originate from the particle itself, so that further ion charging of the

dependent on ion mobility or ion density These latter quantities affect only the rate ofparticle charging [15,24]

For Q < Qsat, it can be shown via surface integration of the field equation, Eq (2.32),that the ion current to the particle is given by

where N is the ambient ion density, qionthe ion charge, and  the ion mobility Solving thisdifferential equation results in an expression for Q as a function of time:

particle charging will be governed by the hyperbolic charging time constant  ¼ 1.1 ms.Note that this latter value is independent of particle radius and electric field magnitude

A charged, airborne particle will experience two principal forces: electrostatic andaerodynamic The former will be given by

where Q is the particle charge, while the latter will be given by the Stokes’ dragequation [9,15]:

(2.38) is valid for particles in the approximate size range 0.5 to 25 mm, for which inertia canusually be ignored For smaller particles, Brownian motion becomes the dominantmechanical force, whereas for particles larger than about 25 mm, the Reynolds number for