Fundamentals of Engineering Electromagnetics - Chapter 2 pdf

2 Applied Electrostatics Mark N. Horenstein Boston University 2.1. INTRODUCTION The term electrostatics brings visions of Benjamin Franklin, the ‘‘kite and key’’ experiment, Leyden jars, cat fur, and glass rods. These and similar experiments heralded the discovery of electromagnetism and were among some of the first recorded in the industrial age. The forces attributable to electrostatic charge have been known since the time of the ancient Greeks, yet the discipline con tinues to be the focus of much research and development. Most electrostatic processes fall into one of two categories. Sometimes, electrostatic charge produces a desired outcome, such as motion, adhesion, or energy dissipation. Electrostatic forces enable such diverse processes as laser printing, electrophotography, electrostatic paint spraying, powder coating, environmentally friendly pesticide application, drug delivery, food production, and electrostatic precipita- tion. Electrostatics is critical to the operation of micro-electromechanical systems (MEMS), including numerous microsensors, transducers, accelerometers, and the and have changed the way electronic circuits interface with the mechanical world. Electrostatic forces on a molecular scale lie at the core of nanodevices, and the inner workings of a cell’s nucleus are also governed by electrostatics. A myriad of self- assembling nanodevices involving coulombic attraction and repulsion comprise yet another technology in which electrostatics plays an important role. Despite its many useful applications, electrostatic charge is often a nuisance to be avoided. For example, sparks of electrostatic origin trigger countless accidental explosions every year and lead to loss of life and property. Less dramatically, static sparks can damage manufactured products such as electronic circuits, photographic film, and thin- coated materials. The transient voltage and current of a single spark event, called an electrostatic discharge (ESD), can render a semiconductor chip useless. Ind eed, a billion- dollar industry specializing in the prevention or neutralization of ESD-producing electrostatic charge has of necessity evolved within the semiconductor industry to help mitigate 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 © 2006 by Taylor & Francis Group, LLC charge can cause webs to cling to rollers and jam production lines. Similarly, the sparks that result from accumulated charge can damage the product itself, either by exposing light-sensitive surfaces or by puncturing the body of the web. This chapter presents the fundamentals that one needs in order to understand electrostatics as both friend and foe. We first define the electrostatic regime in the broad context of Maxwell’s equations and review several fundamental concepts, including Coulomb’s law, force-energy relations, triboelectrification, induction charging, particle electrification, and dielectric breakdown. We then examine several applications of electrostatics in science and industry and discuss some of the methods used to moderate the effects of unwanted charge. 2.2. THE ELECTR OQUASISTATIC REGIME Like all of electromagnetics, electrostatics is governed by Maxwell’s equations, the elegant mathematical statements that form the basis for all that is covered in this book. True electrostatic systems are those in which all time derivatives in Maxwell’s equations are exactly zero and in which forces of magnetic origin are absent. This limiting definition excludes numerou s practical electrostatic-based applications. Fortunately, it can be relaxed while still capturing the salient features of the electrostatic domain. The electroquasistatic regime thus refers to those cases of Maxwell’s equations in which fields and charge magnitudes may vary with time but in which the forces due to the electric field always dominate over the forces due to the magnetic field. At any given moment in time, an electroquasistatic field is identical to the field that would be produced were the relevant charges fixed at their instantaneous values and locations. In order for a system to be electroquasistatic, two conditions must be true: First, any currents that flow within the system must be so small that the magnetic fields they produce generate negligible forces compared to coulombic forces. Second, any time variations in the electric field (or the charges that produce them) must occur so slowly that the effects of any induced magnetic fields are negligible. In this limit, the curl of E approaches zero, and the cross-coupling between E and H that would otherwise give rise to propagating waves is negligible. Thus one manifestation of the electroquasistatic regime is that the so urces of the electric field produce no propagating waves. The conditions for satisfying the electroquasi static limit also can be quantified via dimensional analysis. The curl operator r has the dimensions of a reciprocal distance ÁL, while each time derivative dt in Maxwell’s equations has the dimensions of a time Át. Thus, considering Faraday’s law: rÂE ¼ À@H @t ð2:1Þ the condition that the left-hand side be much greater than the right-hand side becomes dimensionally equivalent to E ÁL ) H Át ð2:2Þ 54 Horenstein © 2006 by Taylor & Francis Group, LLC This same dimensional argument can be applied to Ampere’s law: rÂH ¼ @"E @t þ J ð2:3Þ which, with J ¼0, leads to H ÁL ) "E Át ð2:4Þ Equation (2.4) for H can be substituted into Eq. (2.2), yielding E ÁL )  Át "EÁL Át ð2:5Þ This last equation results in the dimensional condition that ÁL ( Át ffiffiffiffiffiffi " p ð2:6Þ The quantity 1= ffiffiffiffiffiffi " p is the propagation velocity of electromagnetic waves in the medium (i.e., the speed of light), henc e Át= ffiffiffiffiffiffi " p is the distance that a wave would travel after propagating for time Át. If we interpret Át as the period T of a possible propagating wave, then according to Eq. (2.6), the quasistatic limit applies if the length scale Á L of the system 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 Eq. (2.1) becomes rÂE ¼ 0 ð2:7Þ This equation, together with Gauss’ law rÁ"E ¼  ð2:8Þ form the foundations of the electrostatic regime. These two equations can also be expressed in integral form as: þ E Á dl ¼ 0 ð2:9Þ and ð "E Á dA ¼ ð dV ð2:10Þ Applied Electrostatics 55 © 2006 by Taylor & Francis Group, LLC The curl-free electric field Eq. (2.7) can be expressed as the gradient of a scalar potential È: E ¼ÀrÈ ð2:11Þ which can be integrated with respect to path length to yield the definition of the voltage difference between two points a and b: V ab ¼À ð a b E Ádl ð2:12Þ Equation (2.12) applies in any geometry, but it becomes particularly simple for parallel- electrode geometry. For examp le, the two-electrode system of Fig. 2.1, with separation distance d, will produce a uniform electric field of magnitude E y ¼ V d ð2:13Þ when energized to a voltage V. Applying Gauss’ law to the inner surface of the eithe r electrode yields a relationship between the surface charge  s and E y , "E y ¼  s ð2:14Þ Here  s has the units of coulombs per square meter, and " is the dielectric permittivity of the medium between the electrodes. In other, more complex geometries, the solutions to Eqs. (2.9) and (2.10) take on different forms, as discussed in the next section. 2.3. DISCRETE AND DISTRIBUTED CAPACITANCE When two conductors are connected to a voltage source, one will acquire positive charge and the other an equal magnitude of ne gative charge. The charge per unit voltage is called the capacitance of the electrode syst em and can be described by the relationship C ¼ Q V ð2:15Þ Figure 2.1 A simple system consisting of two parallel electrodes of area A separated by a distance d. 56 Horenstein © 2006 by Taylor & Francis Group, LLC Here ÆQ are the magnitudes of the positive and negative charges, and V is the voltage applied to the conductors. It is easily shown that the capacitance between two parallel plane electrodes of area A and separation d is given ap proximately by C ¼ "A d ð2:16Þ where " is the permittivity of the material between the electrod es, and the approximation results because field enhancements, or ‘‘fringing effects,’’ at the edges of the electrodes have been ignored. Although Eq. (2.16) is limited to planar electrodes, it illustrates the following basic form of the formula for capacitance in any geometry: Capacitance ¼ permittivity Âarea parameter length parameter ð2:17Þ a of the field, potential, and capacitance equati ons for 2.4. DIELECTRIC PERMITTIVITY The dielectric permittivity of a material describes its tendency to become internally polarized when subjected to an electric field. Permittivity in farads per meter can also can be expressed in fundamental units of coulombs per volt-meter (C/VÁm). The dielectric constant,orrelative permittivity, of a substance is defined as its permittivity normalized to " 0 , where " 0 ¼8.85 Â10 À12 F/m is the permittivity of free space. For reference purposes, that no material has a permittivity smaller than " 0 . 2.5. THE ORIGINS OF ELECTROSTATIC CHARGE The source of electrostatic charge lies at the atomic level, where a nucleus having a fixed number of positive protons is surrounded by a cloud of orbiting electrons. The number of protons 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, particular ly those in outer orbitals, are easily removed from individual atoms. In conductors such as copper, aluminum, or gold, the outer elect rons are weakly bound to the atom and are free to roam about the crystalline matrix that makes up the material. These free electrons can readily contribute to the flow of electricity. In insulators such as plastics, wood, glass, and ceramics, the outer electrons remain bound to individual atoms, and virtually none are free to contribute to the flow of electricity. Electrostatic phenomena become important when an imbalance exists between positive and negative charges in some region of interest. Sometimes such an imbalance occurs due to the phenomenon of contact electrification [1–8]. When dissimilar materials come into contact and are then separated, one material tends to retain more electrons and become negatively charged, while the other gives up electrons and become positivel y charged. This contact electrification phenomenon, call ed triboelectrification, occurs at the Applied Electrostatics 57 © 2006 by Taylor & Francis Group, LLC Table provides summary energized electrodes in several different geometries. 2.1 relative permittivity values for several common materials are provided in Table 2.2. Note points of intimate material contact. The amount of charge transferred to any given contact point is related to the work function of the materials. The process is enhanced by friction which increases the net contact surface area. Charge separation occurs on both conductors and insulators, but in the former case it becomes significant only when at least one of the conductors is electrically isolated and able to retain the separated charge. This situation is commonly encountered, for example, in the handling of conducting powders. If neither conductor is isolated, an electrical pathway will exist between them, and the separated charges will flow together and neutralize one another. In the case of insulators, however, the separated charges can not easily flow, and the surfaces of the separated objects remain charged. The widespread use of insulators such as plastics and ceramics in industry and manufacturing 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 Geometry E field Potential Capacitance Planar E y ¼ V d È ¼ V y d C ¼ "A d Cylindrical E r ¼ V r lnðb=aÞ È ¼ V lnðb=aÞ ln b r  C ¼ 2"h lnðb=aÞ Spherical E r ¼ V r 2 1=a À 1=b½ È ¼ V a r ðb À rÞ ðb À aÞ C ¼ 4" ba b À a Wedge E  ¼ V r È ¼ V   C ¼ "h  ln b a  Parallel lines (at ÆV ) È % 2"V ln d=aðÞ ln r 1 r 2  r 1 ; r 2 ¼distances to lines C % "h ln d=aðÞ Wire to plane C % 2" cosh À1 ðh þ aÞ=a½ h ) a 58 Horenstein © 2006 by Taylor & Francis Group, LLC The relative propensity of materials to become charged following contact and separation 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 material that 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 negati vely charged. The vagueness of the phrase ‘‘will tend to’’ in the previous sentence is intentional. Despite the seemingly reliable order implied by the triboelectric series, the polarities of tribocharged materials often cannot be predicted reliably, particularly if the materials lie near e ach other in the series. This imprecision is evident in the various sources [9–13] cited in Table 2.3 that differ on the exact order of the series. Contact charging is an imprecise science that is driven by effects Table 2.2 Relative Permittivities of Various Materials Air 1 Polycarbonate $3.0 Alumina 8.8 Polyethylene 2.3 Barium titanate (BaTiO 3 ) 1200 Polyamide $3.4–4.5 Borosilicate glass 4 Polystyrene 2.6 Carbon tetrachloride 2.2 Polyvinyl chloride 6.1 Epoxy $3.4–3.7 Porcelain $5–8 Ethanol 24 Quartz 3.8 Fused quartz (SiO 2 ) 3.9 Rubber $2–4 Gallium arsenide 13.1 Selenium 6 Glass $4–9 Silicon 11.9 Kevlar $3.5–4.5 Silicon nitride 7.2 Methanol 33 Silicone $3.2–4.7 Mylar 3.2 Sodium chloride 5.9 Neoprene $4–6.7 Styrofoam 1.03 Nylon $3.5–4.5 Teflon 2.1 Paper $1.5–3 Water $80 Paraffin 2.1 Wood (dry) 1.4–2.9 Plexiglas 2.8 Table 2.3 The Triboelectric Series POSITIVE Quartz Copper Silicone Zinc Glass Gold Wool Polyester Polymethyl methacrylate (Plexiglas) Polystyrene Salt (NaCl) Natural rubber Fur Polyurethane Silk Polystyrene Aluminum Polyethylene Cellulose acetate Polypropylene Cotton Polyvinyl chloride Steel Silicon Wood Teflon Hard rubber NEGATIVE Source: Compiled from several sources [9–13]. Applied Electrostatics 59 © 2006 by Taylor & Francis Group, LLC occurring on an atomic scale. The slightest trace of surface impurities or altered surface states can cause a material to deviate from the predictions implied by the triboelectric series. Two contact events that seem similar on the macroscopic level can yield entirely different 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 are probabilistic prediction of polarity during multiple charge separation events. Only when two mate rials are located at extremes of the series can their polarities be predicted reliably following a contact-charging event. 2.6. WHEN IS ‘‘STATIC’’ CHARGE TRULY STATIC? The term static electricity invokes an image of charge that cannot flow because it is held stationary 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 considered insulators, however, depends on one’s point of view. Those who work with electrostatic s know that the arrival of a cold, dry winter is synonymous with the onset of ‘‘static season,’’ because electrostatic-related problems are exacerbated by a lack of humidity. When cold air enters a building and is war med, its relative humidity declines noticeably. The tendency of hydroscopic surfaces to absorb moisture, thereby increasing their surface conductivities, is sharply curtailed, and the decay of triboelectric charges to ground over surface-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 to surface 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. Stated succinctly, 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 time constant—the time it takes for separated charges to recombine by flowing over conducting pathways. This relaxation time, be it measured in seconds, hours, or days, must always be compared to time intervals of interest in any given situation. 2.7. INDUCTION CHARGING As discussed in the previous section, contact electrification can result in the separation of charge between two dissimilar materials. Another form of charge separation occurs when a voltage is applied between two conductors, for example the electrodes of a capacitor. Capacitive structures obey the relationship Q ¼ÆCV ð2:18Þ 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 lower potential will carry ÀQ . The mode of charge separation inherent to capacitive structures is known as inductive charging. As Eq. (2.18) suggests, the magnitude of the inductively separated charge can be controlled by altering either C or V. This feature of induction 60 Horenstein © 2006 by Taylor & Francis Group, LLC microscopically dissimilar. The triboelectric series of Table 2.3 should be viewed as a charging lies in contrast to triboelectrification, where the degree of charge separation often depends more on chance than on mechanisms that can be controlled. If a conductor charged by induction is subsequently disconnected from its source of voltage, the now electrically floating conductor will retain its acquired charge regardless of its position relative to other conductors. This mode of induction charging is used often in industry to charge atomized droplets of conducting liquids. The sequence of diagrams shown in Fig. 2.2 illustrates the process. The dispensed liquid becomes part the capacitive electrode as it emerges from the hollow tube and is charged by induction. As the droplet breaks off, it retains its charge, thereafter becoming a free, charged droplet. A droplet of a given size can be charged only to the maximum Raleigh limit [9,14,15]: Q max ¼ 8 ffiffiffiffiffiffiffi " 0  p R 3=2 p ð2:19Þ Here  is the liquid’s surface tension and R p the droplet radius. The Raleigh limit signi fies the value at which self repulsion of the charge overcomes the surface tension holding the droplet together, causing the droplet to break up. 2.8. DIELECTRIC BREAKDOWN Nature is fundamentally charge neutral, but when charges are separated by any mechanism, the maximum quantity of charge is limited by the phenomenon of dielectric breakdown. Dielectric breakdown occurs in solids, liquids and gases and is characterized by the maximum field magnitude that can be sustained before a field-stressed material loses 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), it retains the charge induced on it by the opposing electrode. *Breakdown in vacuum invariably occurs over the surfaces of insulating structures used to support opposing electrodes. Applied Electrostatics 61 © 2006 by Taylor & Francis Group, LLC 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 which current can flow, often with dramatic results. In air and other gases, ever-present stray electrons (produced randomly, for example, by ionizing cosmic rays) will accelerate in an elect ric field, sometimes gaining sufficient energy between collisions to ionize neutral molecules, thereby liberating more electrons. If the field is of sufficient magnitude, the sequence of ensuing collisions can grow exponentially in a self- sustaining avalanche process. Once enough electrons have been liberated from their molecules, the gas becomes locally conducting, resulting in a spark discharge. This phenomenon is familiar to anyone who has walked across a carpet on a dry day and then touched a doorknob or light switch. The human body, having become electrified with excess 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 most dramatic manifestation of this type of discharge is the phenomenon of atmospheric lightning. A good rule of thumb is that air at standard temperature and pressure will break down at a field magnitude of about 30 kV/cm (i.e., 3 MV/m or 3 Â10 6 V/m). This number increases substantially for small air gaps of 50 mm or less because the gap distance approaches the mean free path for collisions, and fewer ionizing events take place. Hence a larger field is required to cause enough ionization to initiate an avalanche breakdown. This phenomena, known as the Paschen effect, results in a breakdown-field versus gap-distance curve such as the one shown in Fig. 2.3 [9,10,12,18,19]. The Paschen effect is critical to the operation 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 made from silicon or other materials. Figure 2.3 Paschen breakdown field vs. gap spacing for air at 1 atmosphere. For large gap spacings, the curve is asymptotic to 3 Â10 6 V/m. 62 Horenstein © 2006 by Taylor & Francis Group, LLC [...]... disconnected from their source of voltage, the charge will thereafter remain constant The stored electrical energy can then be expressed as [24 ] We ¼ Q2 2C 2: 28Þ The force between the electrodes can be found by taking the x derivative of this equation: FQ ¼ dWe Q2 d  x  Q2 ¼ ¼ dx 2 dx "A 2" A 2: 29Þ Equation (2. 29) also describes the force between two insulating surfaces of area A that carry uniform... the order of 100 to 300 mm This large aspect ratio allows the actuator to be modeled by the simple two-electrode capacitive structure shown in Fig 2. 22 © 20 06 by Taylor & Francis Group, LLC 82 Horenstein The electrostatic force in the y direction can be found by taking the derivative of the stored energy (see Sec 2. 10): FE ¼ @ 1 "0 AV 2 CV 2 ¼ @y 2 ðg À y 2 2: 54Þ Here y is the deflection of the bridge,... York, 1958, 25 2 28 1 ´ Tobazeon, R Electrical phenomena of dielectric materials In Handbook of Electrostatic Processes; Chang, J.S., Kelly, A.J., Crowley, J.M., Eds.; Marcel Dekker: New York, 1995; 51– 82 Peek, F.W Dielectric Phenomena in High Voltage Engineering; McGraw-Hill: New York, 1 929 , 48–108 Crowley, J.M Fundamentals of Applied Electrostatics; Wiley: New York, 1986, 164, 20 7 22 5 © 20 06 by Taylor... electrostatic charging of powders J Electrostatics 1985, 16, 175–181 Pratt, T.H Electrostatic Ignitions of Fires and Explosions; Burgoyne: Marietta, GA, 1997, 115–1 52 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 © 20 06 by Taylor & Francis Group, LLC Applied Electrostatics 50 51 52 53 54 55 56 57 58 59 87 Luttgens, G.; Wilson, N Electrostatic Hazards; Butterworth-Heinemann:... systems The DMD is an array of electrostatically-actuated micromirrors of the type shown in Fig 2. 24 Each actuator is capable of being driven into one of two bi-stable positions When voltage is applied to the right-hand pad, as in Fig 2. 24a, the actuator is bent to the right until it reaches its mechanical limit Alternatively, when voltage is applied to the left-hand pad, as in Fig 2. 24b, the actuator bends... setting Er in Eq (2. 32) to zero, yielding Qsat ¼ 3Eo 4%"0 R2 p 2: 33Þ Qsat ¼ 12% "0 R2 Eo p 2: 34Þ or The value given by Eq (2. 34) is called the saturation charge of the particle, or sometimes the Pauthenier limit [27 ] It represents the maximum charge that the particle can hold For a 100-mm particle situated in a 100-kV/m field, for example, the saturation charge calculated from Eq (2. 34) becomes 0.33... A simple actuator is shown here © 20 06 by Taylor & Francis Group, LLC Applied Electrostatics 81 Figure 2. 21 Applying a voltage to the actuator causes the membrane structure to deflect toward the substrate The drawing is not to scale; typical width-to-gap spacing ratios are on the order of 100 Figure 2. 22 The MEMS actuator of Fig 2. 21 can be modeled by the simple mass-spring structure shown here Fe is... will be given by [44] Q1 ¼ C1 V1 þ CM ðV1 À V2 Þ 2: 51Þ The feedback loop of the meter will raise the potential of the probe until V2 ¼ V1, so that Eq (2. 51) becomes V1 ¼ Q1 C1 2: 52 This unambiguous result reflects the potential of the floating conductor with the probe absent One of the more common uses of noncontacting voltmeters involves the measurement of charge on insulating surfaces If surface... that FQ ¼ dWm dx 2: 26Þ As an example of this principle, consider the parallel-electrode structure of Fig 2. 7, for which the capacitance is given by C¼ "A x 2: 27Þ Figure 2. 6 One charged object is displaced relative to another The increment of work added to the system is equal the electrostatic force FQ times the displacement dx © 20 06 by Taylor & Francis Group, LLC 66 Horenstein Figure 2. 7 Parallel electrodes... form of the equation illustrates the significance of the charge-to-mass ratio of the droplet For a given electric field magnitude, the droplet velocity will be proportional to Q/M Because Q has a maximum value determined by either the Raleigh limit of Eq (2. 19) or the saturation charge limit of Eq (2. 34), Eq (2. 42) will be limited as well For a 100-mm droplet of unity density charged to its saturation limit . taking the x derivative of this equation: F Q ¼ dW e dx ¼ Q 2 2 d dx x "A  ¼ Q 2 2"A 2: 29Þ Equation (2. 29) also describes the force between two insulating surfaces of area A that carry. fundamental principle of physics: f 12 ¼ q 1 q 2 4"r 2 2: 24Þ Figure 2. 5 Plot of corona onset voltage V c vs. inner conductor radius a for coaxial electrodes with 10-cm outer conductor radius. 64. applied to the parallel-electrode structure of Fig. 2. 7 with the switch closed. The force be tween the conductors becomes dW e dx ¼ V 2 2 d dx "A x  ¼À "AV 2 2x 2 2: 31Þ This force is