Module 10: Sections 5.1 through 5.4 Module 11: Sections 5.5 through Table of Contents

Transcription

1 Module 1: Sections 5.1 though 5.4 Module 11: Sections 5.5 though Table of Contents 5.1 Intoduction Calculation of Capacitance Example 5.1: Paallel-Plate Capacito Paallel-Plate Capacito Inteactive Simulation Example 5.2: Cylindical Capacito Example 5.3: Spheical Capacito Capacitos in Electic Cicuits Paallel Connection Seies Connection Example 5.4: Equivalent Capacitance Stoing Enegy in a Capacito Enegy Density of the Electic Field Chage Placed between Capacito Plates Inteactive Simulation Example 5.5: Electic Enegy Density of Dy Ai Example 5.6: Enegy Stoed in a Spheical Shell Dielectics Polaization Dielectics without Battey Dielectics with Battey Gauss s Law fo Dielectics Example 5.7: Capacitance with Dielectics Ceating Electic Fields Ceating an Electic Dipole Animation Ceating and Destoying Electic Enegy Animation Summay Appendix: Electic Fields Hold Atoms Togethe Ionic and van de Waals Foces Collection of Chages in Two Dimensions Inteactive Simulation These notes ae excepted Intoduction to Electicity and Magnetism by Sen-Ben Liao, Pete Doumashkin, and John Belche, Copyight 24, ISBN

2 5.8.3 Collection of Chages in Thee Dimensions Inteactive Simulation Collection of Dipoles in Two Dimensions Inteactive Simulation Chaged Paticle Tap Inteactive Simulation Lattice 3D Inteactive Simulation D Electostatic Suspension Bidge Inteactive Simulation D Electostatic Suspension Bidge Inteactive Simulation Poblem-Solving Stategy: Calculating Capacitance Solved Poblems Equivalent Capacitance Capacito Filled with Two Diffeent Dielectics Capacito with Dielectics Capacito Connected to a Sping Conceptual Questions

3 Capacitance and Dielectics 5.1 Intoduction A capacito is a device which stoes electic chage. Capacitos vay in shape and size, but the basic configuation is two conductos caying equal but opposite chages (Figue 5.1.1). Capacitos have many impotant applications in electonics. Some examples include stoing electic potential enegy, delaying voltage changes when coupled with esistos, filteing out unwanted fequency signals, foming esonant cicuits and making fequency-dependent and independent voltage divides when combined with esistos. Some of these applications will be discussed in latte chaptes. Figue Basic configuation of a capacito. In the unchaged state, the chage on eithe one of the conductos in the capacito is zeo. Duing the chaging pocess, a chage Q is moved fom one conducto to the othe one, giving one conducto a chage +Q, and the othe one a chage Q. A potential diffeence ΔV is ceated, with the positively chaged conducto at a highe potential than the negatively chaged conducto. Note that whethe chaged o unchaged, the net chage on the capacito as a whole is zeo. The simplest example of a capacito consists of two conducting plates of aea A, which ae paallel to each othe, and sepaated by a distance d, as shown in Figue Figue A paallel-plate capacito Expeiments show that the amount of chage Q stoed in a capacito is linealy popotional to ΔV, the electic potential diffeence between the plates. Thus, we may wite Q= C ΔV (5.1.1) 5-3

4 whee C is a positive popotionality constant called capacitance. Physically, capacitance is a measue of the capacity of stoing electic chage fo a given potential diffeence ΔV. The SI unit of capacitance is the faad (F) : 1 F = 1 faad = 1 coulomb volt = 1 C V A typical capacitance is in the picofaad ( 1 pf = 1 12 F ) to millifaad ange, (1 mf = 1 3 F=1 μf; 1 μf = 1 6 F ). Figue 5.1.3(a) shows the symbol which is used to epesent capacitos in cicuits. Fo a polaized fixed capacito which has a definite polaity, Figue 5.1.3(b) is sometimes used. (a) (b) 5.2 Calculation of Capacitance Figue Capacito symbols. Let s see how capacitance can be computed in systems with simple geomety. Example 5.1: Paallel-Plate Capacito Conside two metallic plates of equal aea A sepaated by a distance d, as shown in Figue below. The top plate caies a chage +Q while the bottom plate caies a chage Q. The chaging of the plates can be accomplished by means of a battey which poduces a potential diffeence. Find the capacitance of the system. Solution: Figue The electic field between the plates of a paallel-plate capacito To find the capacitance C, we fist need to know the electic field between the plates. A eal capacito is finite in size. Thus, the electic field lines at the edge of the plates ae not staight lines, and the field is not contained entiely between the plates. This is known as 5-4

5 edge effects, and the non-unifom fields nea the edge ae called the finging fields. In Figue the field lines ae dawn by taking into consideation edge effects. Howeve, in what follows, we shall ignoe such effects and assume an idealized situation, whee field lines between the plates ae staight lines. In the limit whee the plates ae infinitely lage, the system has plana symmety and we can calculate the electic field eveywhee using Gauss s law given in Eq. (4.2.5): Ò u u q E d A = enc ε S By choosing a Gaussian pillbox with cap aea A to enclose the chage on the positive plate (see Figue 5.2.2), the electic field in the egion between the plates is EA' = q enc = σ A' E = σ (5.2.1) ε ε ε The same esult has also been obtained in Section using supeposition pinciple. Figue Gaussian suface fo calculating the electic field between the plates. The potential diffeence between the plates is d s = Ed Δ V = V V + = E (5.2.2) + whee we have taken the path of integation to be a staight line fom the positive plate to the negative plate following the field lines (Figue 5.2.2). Since the electic field lines ae always diected fom highe potential to lowe potential, V < V +. Howeve, in computing the capacitance C, the elevant quantity is the magnitude of the potential diffeence: Δ V = Ed (5.2.3) and its sign is immateial. Fom the definition of capacitance, we have 5-5

6 Q C = = ε A (paallel plate ) (5.2.4) Δ V d Note that C depends only on the geometic factos A and d. The capacitance C inceases linealy with the aea A since fo a given potential diffeence ΔV, a bigge plate can hold moe chage. On the othe hand, C is invesely popotional to d, the distance of sepaation because the smalle the value of d, the smalle the potential diffeence ΔV fo a fixed Q Paallel-Plate Capacito Inteactive Simulation This simulation shown in Figue illustates the inteaction of chaged paticles inside the two plates of a capacito. Figue Chaged paticles inteacting inside the two plates of a capacito (link) Each plate contains twelve chages inteacting via Coulomb foce, whee one plate contains positive chages and the othe contains negative chages. Because of thei mutual epulsion, the paticles in each plate ae compelled to maximize the distance between one anothe, and thus spead themselves evenly aound the oute edge of thei enclosue. Howeve, the paticles in one plate ae attacted to the paticles in the othe, so they attempt to minimize the distance between themselves and thei oppositely chaged coespondents. Thus, they distibute themselves along the suface of thei bounding box closest to the othe plate. Example 5.2: Cylindical Capacito Conside next a solid cylindical conducto of adius a suounded by a coaxial cylindical shell of inne adius b, as shown in Figue The length of both cylindes is L and we take this length to be much lage than b a, the sepaation of the cylindes, so that edge effects can be neglected. The capacito is chaged so that the inne cylinde has chage +Q while the oute shell has a chage Q. What is the capacitance? 5-6

7 (a) (b) Figue (a) A cylindical capacito. (b) End view of the capacito. The electic field is non-vanishing only in the egion a < < b. Solution: To calculate the capacitance, we fist compute the electic field eveywhee. Due to the cylindical symmety of the system, we choose ou Gaussian suface to be a coaxial cylinde with length l < L and adius whee a< < b. Using Gauss s law, we have Ò E u u d A = EA = E ( 2π l ) = λl E = ε λ 2πε S (5.2.5) whee λ = Q/ L is the chage pe unit length. Notice that the electic field is nonvanishing only in the egion a< < b. Fo < a, the enclosed chage is q enc = since any net chage in a conducto must eside on its suface. Similaly, fo > b, the enclosed chage is q enc = λ l λ l = since the Gaussian suface encloses equal but opposite chages fom both conductos. The potential diffeence is given by b λ b d λ b Δ V = V b V a = E d = = ln (5.2.6) a 2πε a 2πε a whee we have chosen the integation path to be along the diection of the electic field lines. As expected, the oute conducto with negative chage has a lowe potential. This gives Q λl 2πε L C = = = (5.2.7) ΔV λ ln( b / a )/2 πε ln( b / a ) Once again, we see that the capacitance C depends only on the geometical factos, L, a and b. 5-7

8 Example 5.3: Spheical Capacito As a thid example, let s conside a spheical capacito which consists of two concentic spheical shells of adii a and b, as shown in Figue The inne shell has a chage +Q unifomly distibuted ove its suface, and the oute shell an equal but opposite chage Q. What is the capacitance of this configuation? Figue (a) spheical capacito with two concentic spheical shells of adii a and b. (b) Gaussian suface fo calculating the electic field. Solution: The electic field is non-vanishing only in the egion a< < b. Using Gauss s law, we obtain o u u Q d = E A= E 4π 2 ) = (5.2.8) ε Ò E A ( S E = 1 Q (5.2.9) 2 4πε o Theefoe, the potential diffeence between the two conducting shells is: Δ V = V V = which yields b a a b Q b d Q 1 1 E d = = = Q b a (5.2.1) 4πε a 2 4πε a b 4πε ab Q ab C = = 4πε ΔV b a (5.2.11) Again, the capacitance C depends only on the physical dimensions, a and b. An isolated conducto (with the second conducto placed at infinity) also has a capacitance. In the limit whee b, the above equation becomes 5-8

9 lim C = lim 4 πε ab a = lim 4 πε = 4πε a (5.2.12) b b b a b a 1 b Thus, fo a single isolated spheical conducto of adius R, the capacitance is C = 4πε R (5.2.13) The above expession can also be obtained by noting that a conducting sphee of adius R with a chage Q unifomly distibuted ove its suface has V = Q/4πε R, using infinity as the efeence point having zeo potential,v ( ) =. This gives Q C = Δ V = Q Q /4 πε R = 4πε R (5.2.14) As expected, the capacitance of an isolated chaged sphee only depends on its geomety, namely, the adius R. 5.3 Capacitos in Electic Cicuits A capacito can be chaged by connecting the plates to the teminals of a battey, which ae maintained at a potential diffeence ΔV called the teminal voltage. Figue Chaging a capacito. The connection esults in shaing the chages between the teminals and the plates. Fo example, the plate that is connected to the (positive) negative teminal will acquie some (positive) negative chage. The shaing causes a momentay eduction of chages on the teminals, and a decease in the teminal voltage. Chemical eactions ae then tiggeed to tansfe moe chage fom one teminal to the othe to compensate fo the loss of chage to the capacito plates, and maintain the teminal voltage at its initial level. The battey could thus be thought of as a chage pump that bings a chage Q fom one plate to the othe. 5-9

10 5.3.1 Paallel Connection Suppose we have two capacitos C 1 with chage Q 1 and C 2 with chage Q 2 that ae connected in paallel, as shown in Figue Figue Capacitos in paallel and an equivalent capacito. The left plates of both capacitos C 1 and C 2 ae connected to the positive teminal of the battey and have the same electic potential as the positive teminal. Similaly, both ight plates ae negatively chaged and have the same potential as the negative teminal. Thus, the potential diffeence ΔV is the same acoss each capacito. This gives Q C 1 = 1 Q, C2 = 2 ΔV ΔV (5.3.1) These two capacitos can be eplaced by a single equivalent capacito Ceq with a total chage Q supplied by the battey. Howeve, since Q is shaed by the two capacitos, we must have Q = Q + Q = C Δ V + C Δ V = (C + C ) ΔV (5.3.2) The equivalent capacitance is then seen to be given by Q C eq = = C1 + C 2 (5.3.3) ΔV Thus, capacitos that ae connected in paallel add. The genealization to any numbe of capacitos is N C = C + C + C +L+ C = C (paallel) (5.3.4) eq N i i=1 5-1

11 5.3.2 Seies Connection Suppose two initially unchaged capacitos C 1 and C 2 ae connected in seies, as shown in Figue A potential diffeence ΔV is then applied acoss both capacitos. The left plate of capacito 1 is connected to the positive teminal of the battey and becomes positively chaged with a chage +Q, while the ight plate of capacito 2 is connected to the negative teminal and becomes negatively chaged with chage Q as electons flow in. What about the inne plates? They wee initially unchaged; now the outside plates each attact an equal and opposite chage. So the ight plate of capacito 1 will acquie a chage Q and the left plate of capacito +Q. Figue Capacitos in seies and an equivalent capacito The potential diffeences acoss capacitosc 1 and C 2 ae Q Q Δ V 1 =, Δ V 2 = (5.3.5) C 1 C 2 espectively. Fom Figue 5.3.3, we see that the total potential diffeence is simply the sum of the two individual potential diffeences: Δ V = Δ V 1 + ΔV 2 (5.3.6) In fact, the total potential diffeence acoss any numbe of capacitos in seies connection is equal to the sum of potential diffeences acoss the individual capacitos. These two capacitos can be eplaced by a single equivalent capacito C eq = Q/ ΔV. Using the fact that the potentials add in seies, Q Q Q = + C C C eq 1 2 and so the equivalent capacitance fo two capacitos in seies becomes = + (5.3.7) C C C eq

12 The genealization to any numbe of capacitos connected in seies is N 1 = + +L + = (seies ) (5.3.8) C eq C 1 C 2 C N i=1 C i Example 5.4: Equivalent Capacitance Find the equivalent capacitance fo the combination of capacitos shown in Figue 5.3.4(a) Solution: Figue (a) Capacitos connected in seies and in paallel Since C 1 and C 2 ae connected in paallel, thei equivalent capacitance C 12 is given by C = C + C Figue (b) and (c) Equivalent cicuits. Now capacito C 12 is in seies with C 3, as seen fom Figue 5.3.4(b). So, the equivalent capacitance C 123 is given by 1 C = C C 3 o C C C C 123 = = C 12 + C 3 C 1 + C + C 2 3 C C ( + ) 5-12

13 5.4 Stoing Enegy in a Capacito As discussed in the intoduction, capacitos can be used to stoed electical enegy. The amount of enegy stoed is equal to the wok done to chage it. Duing the chaging pocess, the battey does wok to emove chages fom one plate and deposit them onto the othe. Figue Wok is done by an extenal agent in binging +dq fom the negative plate and depositing the chage on the positive plate. Let the capacito be initially unchaged. In each plate of the capacito, thee ae many negative and positive chages, but the numbe of negative chages balances the numbe of positive chages, so that thee is no net chage, and theefoe no electic field between the plates. We have a magic bucket and a set of stais fom the bottom plate to the top plate (Figue 5.4.1). We stat out at the bottom plate, fill ou magic bucket with a chage +dq, cay the bucket up the stais and dump the contents of the bucket on the top plate, chaging it up positive to chage +dq. Howeve, in doing so, the bottom plate is now chaged to dq. Having emptied the bucket of chage, we now descend the stais, get anothe bucketful of chage +dq, go back up the stais and dump that chage on the top plate. We then epeat this pocess ove and ove. In this way we build up chage on the capacito, and ceate electic field whee thee was none initially. Suppose the amount of chage on the top plate at some instant is +q, and the potential diffeence between the two plates is Δ V =q/ C. To dump anothe bucket of chage +dq on the top plate, the amount of wok done to ovecome electical epulsion is dw = ΔV dq. If at the end of the chaging pocess, the chage on the top plate is +Q, then the total amount of wok done in this pocess is W = Q dq ΔV = Q q 1 Q 2 dq = (5.4.1) C 2 C This is equal to the electical potential enegy U E of the system: 1 Q U E = = Q ΔV = C ΔV (5.4.2) 2 C

14 5.4.1 Enegy Density of the Electic Field One can think of the enegy stoed in the capacito as being stoed in the electic field itself. In the case of a paallel-plate capacito, with C = ε A/ d and Δ V = Ed, we have U = 1 C ΔV 2 = 1 ε A ( Ed ) E 2 2 d 2 2 = 1 ε E 2 ( ) (5.4.3) Since the quantity Ad epesents the volume between the plates, we can define the electic enegy density as Ad u E = U E = 1 ε E 2 (5.4.4) Volume 2 Note that u E is popotional to the squae of the electic field. Altenatively, one may obtain the enegy stoed in the capacito fom the point of view of extenal wok. Since the plates ae oppositely chaged, foce must be applied to maintain a constant sepaation between them. Fom Eq. (4.4.7), we see that a small patch of chage Δ q = σ ( Δ A) expeiences an attactive foce Δ F =σ 2 ( ΔA)/2 ε. If the total aea of the plate is A, then an extenal agent must exet a foce F ext = σ 2 A /2ε to pull the two plates apat. Since the electic field stength in the egion between the plates is given by E = σ / ε, the extenal foce can be ewitten as ε 2 Fext = E A (5.4.5) 2 Note that F ext is independent of d. The total amount of wok done extenally to sepaate the plates by a distance d is then 2 ε E A W ext = F ds = F d = ext ext d (5.4.6) 2 consistent with Eq. (5.4.3). Since the potential enegy of the system is equal to the wok done by the extenal agent, we haveu E = W ext / Ad = ε E 2 / 2. In addition, we note that the expession fo u E is identical to Eq. (4.4.8) in Chapte 4. Theefoe, the electic enegy density u E can also be intepeted as electostatic pessue P Chage Placed between Capacito Plates Inteactive Simulation This applet shown in Figue is a simulation of an expeiment in which an aluminum sphee sitting on the bottom plate of a capacito is lifted to the top plate by the electostatic foce geneated as the capacito is chaged. We have placed a non 5-14

15 conducting baie just below the uppe plate to pevent the sphee fom touching it and dischaging. Figue Electostatic foce expeienced by an aluminum sphee placed between the plates of a paallel-plate capacito (link). While the sphee is in contact with the bottom plate, the chage density of the bottom of the sphee is the same as that of the lowe plate. Thus, as the capacito is chaged, the chage density on the sphee inceases popotional to the potential diffeence between the plates. In addition, enegy flows in to the egion between the plates as the electic field builds up. This can be seen in the motion of the electic field lines as they move fom the edge to the cente of the capacito. As the potential diffeence between the plates inceases, the sphee feels an inceasing attaction towads the top plate, indicated by the inceasing tension in the field as moe field lines "attach" to it. Eventually this tension is enough to ovecome the downwad foce of gavity, and the sphee is lifted. Once sepaated fom the lowe plate, the sphee chage density no longe inceases, and it feels both an attactive foce towads the uppe plate (whose chage is oughly opposite that of the sphee) and a epulsive foce fom the lowe one (whose chage is oughly equal to that of the sphee). The esult is a net foce upwads. Example 5.5: Electic Enegy Density of Dy Ai The beakdown field stength at which dy ai loses its insulating ability and allows a dischage to pass though is E b = V/m. At this field stength, the electic enegy density is: u E = ε E = (885 C /N m )( 3 1 V/m ) = J/m (5.4.7) Example 5.6: Enegy Stoed in a Spheical Shell Find the enegy stoed in a metallic spheical shell of adius a and chage Q. Solution: 5-15

16 The electic field associated of a spheical shell of adius a is (Example 4.3) Q u > E = 4πε ˆ, a 2, < a (5.4.8) The coesponding enegy density is u E = 1 2 Q ε E 2 = (5.4.9) πε outside the sphee, and zeo inside. Since the electic field is non-vanishing outside the spheical shell, we must integate ove the entie egion of space fom = ato =. In 2 spheical coodinates, with dv = 4π d, we have 2 Q 2 Q E 2 d Q a U = 4π d = = = QV a 2 32π ε 8πε 8πε a 2 (5.4.1) whee V = Q/4πε a is the electic potential on the suface of the shell, with V ( ) =. We can eadily veify that the enegy of the system is equal to the wok done in chaging the sphee. To show this, suppose at some instant the sphee has chage q and is at a potential V = q/4πε a. The wok equied to add an additional chage dq to the system is dw = Vdq. Thus, the total wok is W = dw = Vdq = Q q Q 2 dq = (5.4.11) 4πε a 8πε a 5.5 Dielectics In many capacitos thee is an insulating mateial such as pape o plastic between the plates. Such mateial, called a dielectic, can be used to maintain a physical sepaation of the plates. Since dielectics beak down less eadily than ai, chage leakage can be minimized, especially when high voltage is applied. Expeimentally it was found that capacitance C inceases when the space between the conductos is filled with dielectics. To see how this happens, suppose a capacito has a capacitancec when thee is no mateial between the plates. When a dielectic mateial is inseted to completely fill the space between the plates, the capacitance inceases to C = κ e C (5.5.1) 5-16

17 whee κ e is called the dielectic constant. In the Table below, we show some dielectic mateials with thei dielectic constant. Expeiments indicate that all dielectic mateials have κ e > 1. Note that evey dielectic mateial has a chaacteistic dielectic stength which is the maximum value of electic field befoe beakdown occus and chages begin to flow. Mateial 6 κ e Dielectic stength ( 1 V / m ) Ai Pape Glass Wate 8 The fact that capacitance inceases in the pesence of a dielectic can be explained fom a molecula point of view. We shall show that κ e is a measue of the dielectic esponse to an extenal electic field. Thee ae two types of dielectics. The fist type is pola dielectics, which ae dielectics that have pemanent electic dipole moments. An example of this type of dielectic is wate. Figue Oientations of pola molecules when (a) E = and (b) E. As depicted in Figue 5.5.1, the oientation of pola molecules is andom in the absence u of an extenal field. When an extenal electic field E is pesent, a toque is set up and causes the molecules to align with E. Howeve, the alignment is not complete due to andom themal motion. The aligned molecules then geneate an electic field that is opposite to the applied field but smalle in magnitude. The second type of dielectics is the non-pola dielectics, which ae dielectics that do not possess pemanent electic dipole moment. Electic dipole moments can be induced by placing the mateials in an extenally applied electic field. 5-17

18 Figue Oientations of non-pola molecules when (a) E = and (b) E. Figue illustates the oientation of non-pola molecules with and without an u extenal field E. The induced suface chages on the faces poduces an electic field EP u in the diection opposite to E, leading to E= E + E P, with E < E. Below we show u how the induced electic field EP is calculated Polaization We have shown that dielectic mateials consist of many pemanent o induced electic dipoles. One of the concepts cucial to the undestanding of dielectic mateials is the aveage electic field poduced by many little electic dipoles which ae all aligned. Suppose we have a piece of mateial in the fom of a cylinde with aea A and height h, as shown in Figue 5.5.3, and that it consists of N electic dipoles, each with electic dipole moment p spead unifomly thoughout the volume of the cylinde. Figue A cylinde with unifom dipole distibution. We futhemoe assume fo the moment that all of the electic dipole moments p ae aligned with the axis of the cylinde. Since each electic dipole has its own electic field associated with it, in the absence of any extenal electic field, if we aveage ove all the individual fields poduced by the dipole, what is the aveage electic field just due to the pesence of the aligned dipoles? To answe this question, let us define the polaization vecto P to be the net electic dipole moment vecto pe unit volume: 5-18

19 1 N P = p i (5.5.2) Volume i=1 In the case of ou cylinde, whee all the dipoles ae pefectly aligned, the magnitude of P is equal to and the diection of P is paallel to the aligned dipoles. P = Np (5.5.3) Ah Now, what is the aveage electic field these dipoles poduce? The key to figuing this out is ealizing that the situation shown in Figue 5.5.4(a) is equivalent that shown in Figue 5.5.4(b), whee all the little ± chages associated with the electic dipoles in the inteio of the cylinde ae eplaced with two equivalent chages, ±Q P, on the top and bottom of the cylinde, espectively. Figue (a) A cylinde with unifom dipole distibution. (b) Equivalent chage distibution. The equivalence can be seen by noting that in the inteio of the cylinde, positive chage at the top of any one of the electic dipoles is canceled on aveage by the negative chage of the dipole just above it. The only place whee cancellation does not take place is fo electic dipoles at the top of the cylinde, since thee ae no adjacent dipoles futhe up. Thus the inteio of the cylinde appeas unchaged in an aveage sense (aveaging ove many dipoles), wheeas the top suface of the cylinde appeas to cay a net positive chage. Similaly, the bottom suface of the cylinde will appea to cay a net negative chage. How do we find an expession fo the equivalent chage Q P in tems of quantities we know? The simplest way is to equie that the electic dipole moment Q P poduces, Qh P, is equal to the total electic dipole moment of all the little electic dipoles. This gives Qh = Np, o P Q P = Np (5.5.4) h 5-19

20 To compute the electic field poduced by Q P, we note that the equivalent chage distibution esembles that of a paallel-plate capacito, with an equivalent suface chage densityσ P that is equal to the magnitude of the polaization: σ P = Q P = Np = P (5.5.5) A Ah Note that the SI units of P ae (C m)/m 3, o C/m 2, which is the same as the suface chage density. In geneal if the polaization vecto makes an angle θ with nˆ, the outwad nomal vecto of the suface, the suface chage density would be σ P = Pnˆ = P cos θ (5.5.6) Thus, ou equivalent chage system will poduce an aveage electic field of magnitude E P = P / ε. Since the diection of this electic field is opposite to the diection of P, in vecto notation, we have E = P / ε (5.5.7) P Thus, the aveage electic field of all these dipoles is opposite to the diection of the dipoles themselves. It is impotant to ealize that this is just the aveage field due to all the dipoles. If we go close to any individual dipole, we will see a vey diffeent field. We have assumed hee that all ou electic dipoles ae aligned. In geneal, if these dipoles ae andomly oiented, then the polaization P given in Eq. (5.5.2) will be zeo, and thee will be no aveage field due to thei pesence. If the dipoles have some tendency towad a pefeed oientation, then P, leading to a non-vanishing aveage field E. P Let us now examine the effects of intoducing dielectic mateial into a system. We shall fist assume that the atoms o molecules compising the dielectic mateial have a pemanent electic dipole moment. If left to themselves, these pemanent electic dipoles in a dielectic mateial neve line up spontaneously, so that in the absence of any applied extenal electic field, P = due to the andom alignment of dipoles, and the aveage electic field EP is zeo as well. Howeve, when we place the dielectic mateial in an extenal field E, the dipoles will expeience a toque τ = p E that tends to align the with dipole vectos p E. The effect is a net polaization P paallel to E, and theefoe an aveage electic field of the dipoles E anti-paallel to E P, i.e., that will tend to educe the total electic field stength below E. The total electic field E is the sum of these two fields: 5-2

21 E= E + E P = E P / ε (5.5.8) In most cases, the polaization P is not only in the same diection as E, but also linealy popotional to E (and hence E.) This is easonable because without the extenal field E thee would be no alignment of dipoles and no polaization P. We wite the linea elation between P and E as P = εχe e (5.5.9) whee χ e is called the electic susceptibility. Mateials they obey this elation ae linea dielectics. Combing Eqs. (5.5.8) and (5.5.7) gives whee E = (1 + χ e )E = κ e E (5.5.1) κ e = (1 + χ e ) (5.5.11) is the dielectic constant. The dielectic constant κ e is always geate than one since χ e >. This implies E E = κ < E (5.5.12) Thus, we see that the effect of dielectic mateials is always to decease the electic field below what it would othewise be. e In the case of dielectic mateial whee thee ae no pemanent electic dipoles, a simila effect is obseved because the pesence of an extenal field E induces electic dipole moments in the atoms o molecules. These induced electic dipoles ae paallel to E, again leading to a polaization P paallel to E, and a eduction of the total electic field stength Dielectics without Battey As shown in Figue 5.5.5, a battey with a potential diffeence ΔV acoss its teminals is fist connected to a capacito C, which holds a chage Q = C ΔV. We then disconnect the battey, leaving Q =const. 5-21

22 Figue Inseting a dielectic mateial between the capacito plates while keeping the chage Q constant If we then inset a dielectic between the plates, while keeping the chage constant, expeimentally it is found that the potential diffeence deceases by a facto of κ : e This implies that the capacitance is changed to ΔV V Δ = (5.5.13) κ e C = Q Q = Q = κ e = κ e C (5.5.14) ΔV ΔV / κ e ΔV Thus, we see that the capacitance has inceased by a facto of κ.the electic field within e the dielectic is now E = ΔV = ΔV / κ e = 1 ΔV = E (5.5.15) d d κ e d κ e We see that in the pesence of a dielectic, the electic field deceases by a facto of κ. e Dielectics with Battey Conside a second case whee a battey supplying a potential diffeence ΔV emains connected as the dielectic is inseted. Expeimentally, it is found (fist by Faaday) that the chage on the plates is inceased by a facto κ : e Q = κ e Q (5.5.16) whee Q is the chage on the plates in the absence of any dielectic. Figue Inseting a dielectic mateial between the capacito plates while maintaining a constant potential diffeence ΔV. 5-22

23 The capacitance becomes Q C = = κ Q e = κ e C (5.5.17) ΔV ΔV which is the same as the fist case whee the chage Q is kept constant, but now the chage has inceased Gauss s Law fo Dielectics Conside again a paallel-plate capacito shown in Figue 5.5.7: Figue Gaussian suface in the absence of a dielectic. When no dielectic is pesent, the electic field E in the egion between the plates can be found by using Gauss s law: u Q σ d = E A =, E = E A ε ε S We have see that when a dielectic is inseted (Figue 5.5.8), thee is an induced chage Q P of opposite sign on the suface, and the net chage enclosed by the Gaussian suface is Q Q P. Gauss s law becomes Figue Gaussian suface in the pesence of a dielectic. 5-23

24 o u d = EA = Q Q P (5.5.18) Ò E A ε S Q Q E = P (5.5.19) ε A Howeve, we have just seen that the effect of the dielectic is to weaken the oiginal field E by a facto κ e. Theefoe, E = E Q = = Q Q P (5.5.2) κ κ ε A ε A e e fom which the induced chage Q P can be obtained as In tems of the suface chage density, we have 1 Q P = Q 1 (5.5.21) κ e 1 σ P = σ 1 (5.5.22) κ e Note that in the limit κ e = 1, Q P = which coesponds to the case of no dielectic mateial. Substituting Eq. (5.5.21) into Eq. (5.5.18), we see that Gauss s law with dielectic can be ewitten as whee ε = e Q Q Ò E d A = κε = ε (5.5.23) S e κ ε is called the dielectic pemittivity. Altenatively, we may also wite Ò D A S u whee D = εκ E is called the electic displacement vecto. d = Q (5.5.24) 5-24

25 Example 5.7: Capacitance with Dielectics A non-conducting slab of thickness t, aea A and dielectic constant κ e is inseted into the space between the plates of a paallel-plate capacito with spacing d, chage Q and aea A, as shown in Figue 5.5.9(a). The slab is not necessaily halfway between the capacito plates. What is the capacitance of the system? Figue (a) Capacito with a dielectic. (b) Electic field between the plates. Solution: To find the capacitance C, we fist calculate the potential diffeence ΔV. We have aleady seen that in the absence of a dielectic, the electic field between the plates is given by E = Q / ε A, and E D = E /κ e when a dielectic of dielectic constant κ e is pesent, as shown in Figue 5.5.9(b). The potential can be found by integating the electic field along a staight line fom the top to the bottom plates: ΔV = Edl = Δ V Δ V = E ( d t ) E t = Q Q D ( d t + D ) A ε A ε κ e Q 1 = d t 1 Aε κ e t (5.5.25) whee ΔV D = ED t gives is the potential diffeence between the two faces of the dielectic. This C = Δ Q V = d t ε A 1 1 κ e (5.5.26) It is useful to check the following limits: (i) As t, i.e., the thickness of the dielectic appoaches zeo, we have C = ε A / d = C, which is the expected esult fo no dielectic. (ii) As κ e 1, we again have C ε A/ d = C, and the situation also coespond to the case whee the dielectic is absent. 5-25

26 (iii) In the limit whee t d, the space is filled with dielectic, we have C κε e A/ d = κe C. We also comment that the configuation is equivalent to two capacitos connected in seies, as shown in Figue Figue Equivalent configuation. Using Eq. (5.3.8) fo capacitos connected in seies, the equivalent capacitance is 5.6 Ceating Electic Fields 1 d t t = + (5.5.28) C ε A κeε A Ceating an Electic Dipole Animation Electic fields ae ceated by electic chage. If thee is no electic chage pesent, and thee neve has been any electic chage pesent in the past, then thee would be no electic field anywhee is space. How is electic field ceated and how does it come to fill up space? To answe this, conside the following scenaio in which we go fom the electic field being zeo eveywhee in space to an electic field existing eveywhee in space. Figue Ceating an electic dipole (link). (a) Befoe any chage sepaation. (b) Just afte the chages ae sepaated. (c) A long time afte the chages ae sepaated. 5-26

27 Suppose we have a positive point chage sitting ight on top of a negative electic chage, so that the total chage exactly cancels, and thee is no electic field anywhee in space. Now let us pull these two chages apat slightly, so that they ae sepaated by a small distance. If we allow them to sit at that distance fo a long time, thee will now be a chage imbalance an electic dipole. The dipole will ceate an electic field. Let us see how this ceation of electic field takes place in detail. Figue shows thee fames of an animation of the pocess of sepaating the chages. In Figue 5.6.1(a), thee is no chage sepaation, and the electic field is zeo eveywhee in space. Figue 5.6.1(b) shows what happens just afte the chages ae fist sepaated. An expanding sphee of electic fields is obseved. Figue 5.6.1(c) is a long time afte the chages ae sepaated (that is, they have been at a constant distance fom anothe fo a long time). An electic dipole has been ceated. What does this sequence tell us? The following conclusions can be dawn: (1) It is electic chage that geneates electic field no chage, no field. (2) The electic field does not appea instantaneously in space eveywhee as soon as thee is unbalanced chage the electic field popagates outwad fom its souce at some finite speed. This speed will tun out to be the speed of light, as we shall see late. (3) Afte the chage distibution settles down and becomes stationay, so does the field configuation. The initial field patten associated with the time dependent sepaation of the chage is actually a bust of electic dipole adiation. We etun to the subject of adiation at the end of this couse. Until then, we will neglect adiation fields. The field configuation left behind afte a long time is just the electic dipole patten discussed above. We note that the extenal agent who pulls the chages apat has to do wok to keep them sepaate, since they attact each othe as soon as they stat to sepaate. Theefoe, the extenal wok done is to ovecome the electostatic attaction. In addition, the wok also goes into poviding the enegy caied off by adiation, as well as the enegy needed to set up the final stationay electic field that we see in Figue 5.6.1(c). Figue Ceating the electic fields of two point chages by pulling apat two opposite chages initially on top of one anothe. We atificially teminate the field lines at a fixed distance fom the chages to avoid visual confusion (link) 5-27

28 Finally, we ignoe adiation and complete the pocess of sepaating ou opposite point chages that we began in Figue Figue shows the complete sequence. When we finish and have moved the chages fa apat, we see the chaacteistic adial field in the vicinity of a point chage Ceating and Destoying Electic Enegy Animation Let us look at the pocess of ceating electic enegy in a diffeent context. We ignoe enegy losses due to adiation in this discussion. Figue shows one fame of an animation that illustates the following pocess. Figue Ceating (link) and destoying (link) electic enegy We stat out with five negative electic chages and five positive chages, all at the same point in space. Sine thee is no net chage, thee is no electic field. Now we move one of the positive chages at constant velocity fom its initial position to a distance L away along the hoizontal axis. Afte doing that, we move the second positive chage in the same manne to the position whee the fist positive chage sits. Afte doing that, we continue on with the est of the positive chages in the same manne, until all the positive chages ae sitting a distance L fom thei initial position along the hoizontal axis. Figue shows the field configuation duing this pocess. We have colo coded the gass seeds epesentation to epesent the stength of the electic field. Vey stong fields ae white, vey weak fields ae black, and fields of intemediate stength ae yellow. Ove the couse of the ceate animation associated with Figue 5.6.3, the stength of the electic field gows as each positive chage is moved into place. The electic enegy flows out fom the path along which the chages move, and is being povided by the agent moving the chage against the electic field of the othe chages. The wok that this agent does to sepaate the chages against thei electic attaction appeas as enegy in the electic field. We also have an animation of the opposite pocess linked to Figue That is, we etun in sequence each of the five positive chages to thei oiginal positions. At the end of this pocess we no longe have an electic field, because we no longe have an unbalanced electic chage. On the othe hand, ove the couse of the destoy animation associated with Figue 5.6.3, the stength of the electic field deceases as each positive chage is etuned to its oiginal position. The enegy flows fom the field back to the path along which the 5-28

29 chages move, and is now being povided to the agent moving the chage at constant speed along the electic field of the othe chages. The enegy povided to that agent as we destoy the electic field is exactly the amount of enegy that the agent put into ceating the electic field in the fist place, neglecting adiative losses (such losses ae small if we move the chages at speeds small compaed to the speed of light). This is a totally evesible pocess if we neglect such losses. That is, the amount of enegy the agent puts into ceating the electic field is exactly etuned to that agent as the field is destoyed. Thee is one final point to be made. Wheneve electomagnetic enegy is being ceated, an electic chage is moving (o being moved) against an electic field ( q v E< ). Wheneve electomagnetic enegy is being destoyed, an electic chage is moving (o being moved) along an electic field ( q ve > ). When we etun to the ceation and destuction of magnetic enegy, we will find this ule holds thee as well. 5.7 Summay A capacito is a device that stoes electic chage and potential enegy. The capacitance C of a capacito is the atio of the chage stoed on the capacito plates to the the potential diffeence between them: Q C = ΔV System Capacitance Isolated chaged sphee of adius R C = 4πε R Paallel-plate capacito of plate aea A and plate sepaation d A C = ε d Cylindical capacito of length L, inne adius a and oute adius b 2 πε L C = ln( b/ a ) Spheical capacito with inne adius a and oute adius b C = 4 πε ab ( b a) The equivalent capacitance of capacitos connected in paallel and in seies ae C eq = C 1 + C 2 + C 3 +L (paallel) = + + +L (seies) C eq C 1 C 2 C

30 The wok done in chaging a capacito to a chage Q is U = Q2 = 1 Q ΔV = 1 C ΔV 2 2C 2 2 This is equal to the amount of enegy stoed in the capacito. The electic enegy can also be thought of as stoed in the electic field E. The enegy density (enegy pe unit volume) is 1 u E = 2 ε E 2 The enegy density u E is equal to the electostatic pessue on a suface. When a dielectic mateial with dielectic constant κ is inseted into a e capacito, the capacitance inceases by a facto κ e : C = κ e C The polaization vecto P is the magnetic dipole moment pe unit volume: N 1 P = p V i=1 The induced electic field due to polaization is E P = P / ε i In the pesence of a dielectic with dielectic constant κ e, the electic field becomes E= E + E = E / κ P e whee E is the electic field without dielectic. 5.8 Appendix: Electic Fields Hold Atoms Togethe In this Appendix, we illustate how electic fields ae esponsible fo holding atoms togethe. 5-3

31 As ou mental eye penetates into smalle and smalle distances and shote and shote times, we find natue behaving so entiely diffeently fom what we obseve in visible and palpable bodies of ou suoundings that no model shaped afte ou lage-scale expeiences can eve be "tue". A completely satisfactoy model of this type is not only pactically inaccessible, but not even thinkable. O, to be pecise, we can, of couse, think of it, but howeve we think it, it is wong Ionic and van de Waals Foces Ewin Schoedinge Electomagnetic foces povide the glue that holds atoms togethe that is, that keep electons nea potons and bind atoms togethe in solids. We pesent hee a bief and vey idealized model of how that happens fom a semi-classical point of view. (a) Figue (a) A negative chage (link) and (b) a positive chage (link) move past a massive positive paticle at the oigin and is deflected fom its path by the stesses tansmitted by the electic fields suounding the chages. Figue 5.8.1(a) illustates the examples of the stesses tansmitted by fields, as we have seen befoe. In Figue 5.8.1(a) we have a negative chage moving past a massive positive chage and being deflected towad that chage due to the attaction that the two chages feel. This attaction is mediated by the stesses tansmitted by the electomagnetic field, and the simple intepetation of the inteaction shown in Figue 5.8.1(b) is that the attaction is pimaily due to a tension tansmitted by the electic fields suounding the chages. In Figue 5.8.1(b) we have a positive chage moving past a massive positive chage and being deflected away fom that chage due to the epulsion that the two chages feel. This epulsion is mediated by the stesses tansmitted by the electomagnetic field, as we have discussed above, and the simple intepetation of the inteaction shown in Figue 5.8.1(b) is that the epulsion is pimaily due to a pessue tansmitted by the electic fields suounding the chages. Conside the inteaction of fou chages of equal mass shown in Figue Two of the chages ae positively chaged and two of the chages ae negatively chaged, and all have the same magnitude of chage. The paticles inteact via the Coulomb foce. (b) 5-31

32 We also intoduce a quantum-mechanical Pauli foce, which is always epulsive and becomes vey impotant at small distances, but is negligible at lage distances. The citical distance at which this epulsive foce begins to dominate is about the adius of the sphees shown in Figue This Pauli foce is quantum mechanical in oigin, and keeps the chages fom collapsing into a point (i.e., it keeps a negative paticle and a positive paticle fom sitting exactly on top of one anothe). Additionally, the motion of the paticles is damped by a tem popotional to thei velocity, allowing them to "settle down" into stable (o meta-stable) states. Figue Fou chages inteacting via the Coulomb foce, a epulsive Pauli foce at close distances, with dynamic damping (link) When these chages ae allowed to evolve fom the initial state, the fist thing that happens (vey quickly) is that the chages pai off into dipoles. This is a apid pocess because the Coulomb attaction between unbalanced chages is vey lage. This pocess is called "ionic binding", and is esponsible fo the inte-atomic foces in odinay table salt, NaCl. Afte the dipoles fom, thee is still an inteaction between neighboing dipoles, but this is a much weake inteaction because the electic field of the dipoles falls off much faste than that of a single chage. This is because the net chage of the dipole is zeo. When two opposite chages ae close to one anothe, thei electic fields almost cancel each othe out. Although in pinciple the dipole-dipole inteaction can be eithe epulsive o attactive, in pactice thee is a toque that otates the dipoles so that the dipole-dipole foce is attactive. Afte a long time, this dipole-dipole attaction bings the two dipoles togethe in a bound state. The foce of attaction between two dipoles is temed a van de Waals foce, and it is esponsible fo intemolecula foces that bind some substances togethe into a solid Collection of Chages in Two Dimensions Inteactive Simulation Figue is an inteactive two-dimensional ShockWave display that shows the same dynamical situation as in Figue except that we have included a numbe of positive and negative chages, and we have eliminated the epesentation of the field so that we 5-32

33 can inteact with this simulation in eal time. We stat the chages at est in andom positions in space, and then let them evolve accoding to the foces that act on them (electostatic attaction/epulsion, Pauli epulsion at vey shot distances, and a dynamic dag tem popotional to velocity). The paticles will eventually end up in a configuation in which the net foce on any given paticle is essentially zeo. As we saw in the animation in Figue 5.8.3, geneally the individual paticles fist pai off into dipoles and then slowly combine into lage stuctues. Rings and staight lines ae the most common configuations, but by clicking and dagging paticles aound, the use can coax them into moe complex meta-stable fomations. Figue A two dimensional inteactive simulation of a collection of positive and negative chages affected by the Coulomb foce and the Pauli epulsive foce, with dynamic damping (link) In paticula, ty this sequence of actions with the display. Stat it and wait until the simulation has evolved to the point whee you have a line of paticles made up of seven o eight paticles. Left click on one of the end chages of this line and dag it with the mouse. If you do this slowly enough, the entie line of chaes will follow along with the chage you ae vitually touching. When you move that chage, you ae putting enegy into the chage you have selected on one end of the line. This enegy is going into moving that chage, but it is also being supplied to the est of the chages via thei electomagnetic fields. The enegy that the chage on the opposite end of the line eceives a little while afte you stat moving the fist chage is deliveed to it entiely by enegy flowing though space in the electomagnetic field, fom the site whee you ceate that enegy. This is a micocosm of how you inteact with the wold. A physical object lying on the floo in font is held togethe by electostatic foces. Quantum mechanics keeps it fom collapsing; electostatic foces keep it fom flying apat. When you each down and pick that object up by one end, enegy is tansfeed fom whee you gasp the object to the est of it by enegy flow in the electomagnetic field. When you aise it above the floo, the tail end of the object neve touches the point whee you gasp it. All of the enegy povided to the tail end of the object to move it upwad against gavity is povided by enegy flow via electomagnetic fields, though the complicated web of electomagnetic fields that hold the object togethe. 5-33

34 5.8.3 Collection of Chages in Thee Dimensions Inteactive Simulation Figue is an inteactive thee-dimensional ShockWave display that shows the same dynamical situation as in Figue except that we ae looking at the scene in thee dimensions. This display can be otated to view fom diffeent angles by ight-clicking and dagging in the display. We stat the chages at est in andom positions in space, and then let them evolve accoding to the foces that act on them (electostatic attaction/epulsion, Pauli epulsion at vey shot distances, and a dynamic dag tem popotional to velocity). Hee the configuations ae moe complex because of the availability of the thid dimension. In paticula, one can hit the w key to toggle a foce that pushes the chages togethe on and off. Toggling this foce on and letting the chages settle down in a clump, and then toggling it off to let them expand, allows the constuction of complicated thee dimension stuctues that ae meta-stable. An example of one of these is given in Figue Figue An thee-dimensional inteactive simulation of a collection of positive and negative chages affected by the Coulomb foce and the Pauli epulsive foce, with dynamic damping (link) Collection of Dipoles in Two Dimensions Inteactive Simulation Figue shows an inteactive ShockWave simulation that allows one to inteact in two dimensions with a goup of electic dipoles. Figue An inteactive simulation of a collection of electic dipoles affected by the Coulomb foce and the Pauli epulsive foce, with dynamic damping (link) The dipoles ae ceated with andom positions and oientations, with all the electic dipole vectos in the plane of the display. As we noted above, although in pinciple the 5-34

35 dipole-dipole inteaction can be eithe epulsive o attactive, in pactice thee is a toque that otates the dipoles so that the dipole-dipole foce is attactive. In the ShockWave simulation we see this behavio that is, the dipoles oient themselves so as to attact, and then the attaction gathes them togethe into bound stuctues Chaged Paticle Tap Inteactive Simulation Figue shows an inteactive simulation of a chaged paticle tap. Figue An inteactive simulation of a paticle tap (link). Paticles inteact as befoe, but in addition each paticle feels a foce that pushes them towad the oigin, egadless of the sign of thei chage. That tapping foce inceases linealy with distance fom the oigin. The chages initially ae andomly distibuted in space, but as time inceases the dynamic damping cools the paticles and they cystallize into a numbe of highly symmetic stuctues, depending on the numbe of paticles. This mimics the highly odeed stuctues that we see in natue (e.g., snowflakes). Execise: Stat the simulation. The simulation initially intoduces 12 positive chages in andom positions (you can of couse add moe paticles of eithe sign, but fo the moment we deal with only the initial 12). About half the time, the 12 chages will settle down into an equilibium in which thee is a chage in the cente of a sphee on which the othe 11 chages ae aanged. The othe half of the time all 12 paticles will be aanged on the suface of a sphee, with no chage in the middle. Whicheve aangement you initially find, see if you can move one of the paticles into position so that you get to the othe stable configuation. To move a chage, push shift and left click, and use the aow buttons to move it up, down, left, and ight. You may have to select seveal diffeent chages in tun to find one that you can move into the cente, if you initial equilibium does not have a cente chage. Hee is anothe execise. Put an additional 8 positive chages into the display (by pessing p eight times) fo a total of 2 chages. By moving chages aound as above, you can get two chages in inside a spheical distibution of the othe 18. Is this the 5-35

36 lowest numbe of chages fo which you can get equilibium with two chages inside? That is, can you do this with 18 chages? Note that if you push the s key you will get geneate a suface based on the positions of the chages in the sphee, which will make its symmeties moe appaent Lattice 3D Inteactive Simulation Lattice 3D, shown in Figue 5.8.7, simulates the inteaction of chaged paticles in thee dimensions. The paticles inteact via the classical Coulomb foce, as well as the epulsive quantum-mechanical Pauli foce, which acts at close distances (accounting fo the collisions between them). Additionally, the motion of the paticles is damped by a tem popotional to thei velocity, allowing them to settle down into stable (o metastable) states. Figue Lattice 3D simulating the inteaction of chaged paticles in thee dimensions (link). In this simulation, the popotionality of the Coulomb and Pauli foces has been adjusted to allow fo lattice fomation, as one might see in a cystal. The pefeed stable state is a ectangula (cubic) lattice, although othe fomations ae possible depending on the numbe of paticles and thei initial positions. Selecting a paticle and pessing f will toggle field lines illustating the local field aound that paticle. Pefomance vaies depending on the numbe of paticles / field lines in the simulation D Electostatic Suspension Bidge Inteactive Simulation To connect electostatic foces to one moe example of the eal wold, Figue is a simulation of a 2D electostatic suspension bidge. The bidge is ceated by attaching a seies of positive and negatively chaged paticles to two fixed endpoints, and adding a downwad gavitational foce. The tension in the bidge is supplied simply by the Coulomb inteaction of its constituent pats and the Pauli foce keeps the chages fom collapsing in on each othe. Initially, the bidge only sags slightly unde the weight of gavity. Howeve the use can intoduce additional neutal paticles (by pessing o ) 5-36

37 to stess the bidge moe, until the electostatic bonds beak unde the stess and the bidge collapses. Figue A ShockWave simulation of a 2D electostatic suspension bidge (link) D Electostatic Suspension Bidge Inteactive Simulation In the simulation shown in Figue 5.8.9, a 3D electostatic suspension bidge is ceated by attaching a lattice of positive and negatively chaged paticles between fou fixed cones, and adding a downwad gavitational foce. The tension in the bidge is supplied simply by the Coulomb inteaction of its constituent pats and the Pauli foce keeping them fom collapsing in on each othe. Initially, the bidge only sags slightly unde the weight of gavity, but what would happen to it unde a ain of massive neutal paticles? Pess o to find out. Figue A ShockWave simulation of a 3D electostatic suspension bidge (link). 5.9 Poblem-Solving Stategy: Calculating Capacitance In this chapte, we have seen how capacitance C can be calculated fo vaious systems. The pocedue is summaized below: (1) Identify the diection of the electic field using symmety. (2) Calculate the electic field eveywhee. 5-37

38 (3) Compute the electic potential diffeence ΔV. (4) Calculate the capacitance C using C = Q/ ΔV. In the Table below, we illustate how the above steps ae used to calculate the capacitance of a paallel-plate capacito, cylindical capacito and a spheical capacito. Capacitos Paallel-plate Cylindical Spheical Figue (1) Identify the diection of the electic field using symmety (2) Calculate electic field eveywhee E S u u Q d A = EA = ε Q σ E = = A ε ε u u ( 2 π ) Ò E d A = E l = 2 ε S ( ) Q Ò E d A = E 4 π = ε S λ E = 2 πε Q u u 1 Q E = 2 4 πε o (3) Compute the electic potential diffeence ΔV Δ V = V V + = E d s + = Ed Δ V = V V = E d b a a λ b = ln 2 πε a b Δ V = V V = E d b a a Q b a = 4 πε ab b (4) Calculate C using C = Q / Δ V ε C = d A C 2 πε l ln( b/ a ) = C 4 ab = πε b a 5-38

39 5.1 Solved Poblems Equivalent Capacitance Conside the configuation shown in Figue Find the equivalent capacitance, assuming that all the capacitos have the same capacitance C. Solution: Figue Combination of Capacitos Fo capacitos that ae connected in seies, the equivalent capacitance is 1 = L = 1 (seies) C eq C 1 C 2 i C i On the othe hand, fo capacitos that ae connected in paallel, the equivalent capacitance is C eq = C 1 + C 2 +L = C i i (paallel) Using the above fomula fo seies connection, the equivalent configuation is shown in Figue Figue Now we have thee capacitos connected in paallel. The equivalent capacitance is given by C eq = C 1+ + = C

40 5.1.2 Capacito Filled with Two Diffeent Dielectics Two dielectics with dielectic constants κ 1 and κ 2 each fill half the space between the plates of a paallel-plate capacito as shown in Figue Figue Capacito filled with two diffeent dielectics. Each plate has an aea A and the plates ae sepaated by a distance d. Compute the capacitance of the system. Solution: Since the potential diffeence on each half of the capacito is the same, we may teat the system as being composed of two capacitos connected in paallel. Thus, the capacitance of the system is With we obtain C = C 1 + C 2 i A C i = κε ( /2), i =1,2 d C = κε (A/ 2) + κ ε (A/ 2) = ε A 1 2 (κ 1 +κ 2 ) d d 2d Capacito with Dielectics Conside a conducting spheical shell with an inne adius a and oute adius c. Let the space between two sufaces be filed with two diffeent dielectic mateials so that the dielectic constant is κ 1 between a and b, and κ 2 between b and c, as shown in Figue Detemine the capacitance of this system. 5-4

41 Figue Spheical capacito filled with dielectics. Solution: The system can be teated as two capacitos connected in seies, since the total potential diffeence acoss the capacitos is the sum of potential diffeences acoss individual capacitos. The equivalent capacitance fo a spheical capacito of inne adius 1 and oute adius 2 filled with dielectic with dielectic constant κ e is given by 1 2 C = 4πε κ e 2 1 Thus, the equivalent capacitance of this system is o κ 2 cb ( a) + κ 1 ac ( b) = + = C 4πε κ 1 ab 4πε κ 2bc 4πε κ1κ 2 abc b (b a) (c ) 4πε κ1κ 2 abc C = κ 2 cb ( a) + κ ac b) It is instuctive to check the limit whee κ1, κ2 1. In this case, the above expession educes to 1 ( 4πε abc 4πε abc 4πε ac C = = = cb ( a) + ac ( b) bc ( a) (c a) which agees with Eq. (5.2.11) fo a spheical capacito of inne adius a and oute adius c Capacito Connected to a Sping 5-41