Chapter 6 Current and Resistance
|
|
- Scott Patrick
- 6 years ago
- Views:
Transcription
1 Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt Currnt Dnsity Ohm s Law Elctrical Enrgy and Powr Summary Solvd Problms Rsistivity of a Cabl Charg at a Junction Drift Vlocity Rsistanc of a Truncatd Con Rsistanc of a Hollow Cylindr Concptual Qustions Additional Problms Currnt and Currnt Dnsity Powr Loss and Ohm s Law Rsistanc of a Con Currnt Dnsity and Drift Spd Currnt Sht Rsistanc and Rsistivity Powr, Currnt, and Voltag Charg Accumulation at th Intrfac
2 Currnt and Rsistanc 6.1 Elctric Currnt Elctric currnts ar flows of lctric charg. Suppos a collction of chargs is moving prpndicular to a surfac of ara A, as shown in Figur Figur Chargs moving through a cross sction. Th lctric currnt is dfind to b th rat at which chargs flow across any crosssctional ara. If an amount of charg Q passs through a surfac in a tim intrval t, thn th avrag currnt I avg is givn by Q Iavg = (6.1.1) t Th SI unit of currnt is th ampr (A), with 1 A = 1 coulomb/sc. Common currnts rang from mga-amprs in lightning to nano-amprs in your nrvs. In th limit t 0, th instantanous currnt I may b dfind as I dq = (6.1.) dt Sinc flow has a dirction, w hav implicitly introducd a convntion that th dirction of currnt corrsponds to th dirction in which positiv chargs ar flowing. Th flowing chargs insid wirs ar ngativly chargd lctrons that mov in th opposit dirction of th currnt. Elctric currnts flow in conductors: solids (mtals, smiconductors), liquids (lctrolyts, ionizd) and gass (ionizd), but th flow is impdd in nonconductors or insulators Currnt Dnsity To rlat currnt, a macroscopic quantity, to th microscopic motion of th chargs, lt s xamin a conductor of cross-sctional ara A, as shown in Figur
3 Figur 6.1. A microscopic pictur of currnt flowing in a conductor. Lt th total currnt through a surfac b writtn as I = d A (6.1.3) J whr J is th currnt dnsity (th SI unit of currnt dnsity ar A/m ). If q is th charg of ach carrir, and n is th numbr of charg carrirs pr unit volum, th total amount of charg in this sction is thn Q = q( na x). Suppos that th charg carrirs mov with a spd vd ; thn th displacmnt in a tim intrval t will b x = vd t, which implis I avg Q = = nqvd A t (6.1.4) Th spd at which th charg carrirs ar moving is known as th drift spd. v d Physically, v d is th avrag spd of th charg carrirs insid a conductor whn an xtrnal lctric fild is applid. Actually an lctron insid th conductor dos not travl in a straight lin; instad, its path is rathr rratic, as shown in Figur Figur Motion of an lctron in a conductor. From th abov quations, th currnt dnsity J can b writtn as J = nqv d (6.1.5) Thus, w s that J and v d point in th sam dirction for positiv charg carrirs, in opposit dirctions for ngativ charg carrirs.
4 To find th drift vlocity of th lctrons, w first not that an lctron in th conductor xprincs an lctric forc F = E which givs an acclration a F m = = E m (6.1.6) Lt th vlocity of a givn lctron immdiat aftr a collision b v i. Th vlocity of th lctron immdiatly bfor th nxt collision is thn givn by whr t is th tim travld. Th avrag of f = i + t = i E v v a v t m v f ovr all tim intrvals is (6.1.) v f = i E v m t (6.1.8) which is qual to th drift vlocity v d. Sinc in th absnc of lctric fild, th vlocity of th lctron is compltly random, it follows that v 0. If τ = t is th avrag charactristic tim btwn succssiv collisions (th man fr tim), w hav Th currnt dnsity in Eq. (6.1.5) bcoms v d = = E v f m τ i = E n τ J = nvd = n τ = E m m (6.1.9) (6.1.10) Not that J and E will b in th sam dirction for ithr ngativ or positiv charg carrirs. 6. Ohm s Law In many matrials, th currnt dnsity is linarly dpndnt on th xtrnal lctric fild E. Thir rlation is usually xprssd as J = σ E (6..1) 3
5 whr σ is calld th conductivity of th matrial. Th abov quation is known as th (microscopic) Ohm s law. A matrial that obys this rlation is said to b ohmic; othrwis, th matrial is non-ohmic. Comparing Eq. (6..1) with Eq. (6.1.10), w s that th conductivity can b xprssd as n τ σ = (6..) m To obtain a mor usful form of Ohm s law for practical applications, considr a sgmnt of straight wir of lngth l and cross-sctional ara A, as shown in Figur Figur 6..1 A uniform conductor of lngth l and potntial diffrnc V = V V. Suppos a potntial diffrnc V = Vb V a is applid btwn th nds of th wir, crating an lctric fild E and a currnt I. Assuming E to b uniform, w thn hav b a V = V V = d s = El b a b E a (6..3) Th currnt dnsity can thn b writtn as J V σe σ = = (6..4) l With J = I / A, th potntial diffrnc bcoms l l V = J = I = RI (6..5) σ σ A whr V l R = = (6..6) I σ A is th rsistanc of th conductor. Th quation 4
6 V = IR (6..) is th macroscopic vrsion of th Ohm s law. Th SI unit of R is th ohm (Ω, Grk lttr Omga), whr 1V 1 Ω (6..8) 1A Onc again, a matrial that obys th abov rlation is ohmic, and non-ohmic if th rlation is not obyd. Most mtals, with good conductivity and low rsistivity, ar ohmic. W shall focus mainly on ohmic matrials. Figur 6.. Ohmic vs. Non-ohmic bhavior. Th rsistivity ρ of a matrial is dfind as th rciprocal of conductivity, 1 m ρ = σ = n τ (6..9) From th abov quations, w s that ρ can b rlatd to th rsistanc R of an objct by or E V / l RA ρ = = = J I / A l ρl R = (6..10) A Th rsistivity of a matrial actually varis with tmpratur T. For mtals, th variation is linar ovr a larg rang of T: [ T T ] ρ = ρ0 1 + α( 0 ) (6..11) whr α is th tmpratur cofficint of rsistivity. Typical valus of ρ, σ and α (at 0 C) for diffrnt typs of matrials ar givn in th Tabl blow. 5
7 Matrial Elmnts Silvr Rsistivity ρ ( Ω m) Coppr Aluminum Tungstn Conductivity σ ( Ω m) 1 Tmpratur Cofficint α (C) Iron Platinum Alloys Brass Manganin Nichrom Smiconductors Carbon (graphit) Grmanium (pur) Silicon (pur) Insulators Glass Sulfur Quartz (fusd) Elctrical Enrgy and Powr Considr a circuit consisting of a battry and a rsistor with rsistanc R (Figur 6.3.1). Lt th potntial diffrnc btwn two points a and b b V = Vb Va > 0. If a charg q is movd from a through th battry, its lctric potntial nrgy is incrasd by U = q V. On th othr hand, as th charg movs across th rsistor, th potntial nrgy is dcrasd du to collisions with atoms in th rsistor. If w nglct th intrnal rsistanc of th battry and th conncting wirs, upon rturning to a th potntial nrgy of q rmains unchangd. Figur A circuit consisting of a battry and a rsistor of rsistanc R. 6
8 Thus, th rat of nrgy loss through th rsistor is givn by U q P = = V = I V t t (6.3.1) This is prcisly th powr supplid by th battry. Using V = IR, on may rwrit th abov quation as ( V ) P= I R= (6.3.) R 6.4 Summary Th lctric currnt is dfind as: I dq = dt Th avrag currnt in a conductor is I avg = nqv A d whr n is th numbr dnsity of th charg carrirs, q is th charg ach carrir has, v is th drift spd, and A is th cross-sctional ara. d Th currnt dnsity J through th cross sctional ara of th wir is J = nqv Microscopic Ohm s law: th currnt dnsity is proportional to th lctric fild, and th constant of proportionality is calld conductivity σ : d J = σ E Th rciprocal of conductivity σ is calld rsistivity ρ : 1 ρ = σ Macroscopic Ohm s law: Th rsistanc R of a conductor is th ratio of th potntial diffrnc V btwn th two nds of th conductor and th currnt I:
9 V R = I Rsistanc is rlatd to rsistivity by ρl R = A whr l is th lngth and A is th cross-sctional ara of th conductor. Th drift vlocity of an lctron in th conductor is v d = E m τ whr m is th mass of an lctron, and succssiv collisions. τ is th avrag tim btwn Th rsistivity of a mtal is rlatd to τ by 1 m ρ = = σ n τ Th tmpratur variation of rsistivity of a conductor is ( ) ρ = ρ0 1+ α T T 0 whr α is th tmpratur cofficint of rsistivity. Powr, or rat at which nrgy is dlivrd to th rsistor is 6.5 Solvd Problms P= I V = I R= ( ) V R Rsistivity of a Cabl A 3000-km long cabl consists of svn coppr wirs, ach of diamtr 0.3 mm, bundld togthr and surroundd by an insulating shath. Calculat th rsistanc of th 6 cabl. Us 3 10 Ω cm for th rsistivity of th coppr. 8
10 Solution: Th rsistanc R of a conductor is rlatd to th rsistivity ρ by R = ρl/ A, whr l and A ar th lngth of th conductor and th cross-sctional ara, rspctivly. Sinc th cabl consists of N = coppr wirs, th total cross sctional ara is d (0.03cm) A= Nπ r = N π = π 4 4 Th rsistanc thn bcoms R ρl A 6 8 ( Ω )( ) 3 10 cm 3 10 cm π 0.03cm / 4 4 = = = Ω ( ) 6.5. Charg at a Junction Show that th total amount of charg at th junction of th two matrials in Figur is ε0i( σ σ1 ), whr I is th currnt flowing through th junction, andσ 1 and σ ar th conductivitis for th two matrials. Solution: Figur Charg at a junction. In a stady stat of currnt flow, th normal componnt of th currnt dnsity J must b th sam on both sids of th junction. Sinc J = σ E, w hav σ1e1 = σ E or E σ 1 E = σ 1 Lt th charg on th intrfac b, w hav, from th Gauss s law: q in or E S q = = ( ) in d A E E1 A ε 0 9
11 E E = 1 qin Aε 0 Substituting th xprssion for E from abov thn yilds Sinc th currnt is ( σ ) σ qin = ε0ae1 1 = ε0aσ1 E1 σ σ σ 1 I = JA = E A, th amount of charg on th intrfac bcoms 1 1 q 1 1 = ε I σ σ 1 in Drift Vlocity Th rsistivity of sawatr is about 5 Ω cm. Th charg carrirs ar chifly Na + and Cl ions, and of ach thr ar about / cm. If w fill a plastic tub mtrs long with sawatr and connct a 1-volt battry to th lctrods at ach nd, what is th rsulting avrag drift vlocity of th ions, in cm/s? Solution: Th currnt in a conductor of cross sctional ara A is rlatd to th drift spd charg carrirs by I = nav d v d of th whr n is th numbr of chargs pr unit volum. W can thn rwrit th Ohm s law as ρl V = IR = ( navd) = nvdρl A which yilds v d V = nρl Substituting th valus, w hav v d 1V 5 V cm 5 cm = =.5 10 =.5 10 C Ω s ( 6 10 /cm )( C)( 5Ω cm)( 00cm) 10
12 In convrting th units w hav usd V V 1 A = = = s Ω C Ω C C Rsistanc of a Truncatd Con Considr a matrial of rsistivity ρ in a shap of a truncatd con of altitud h, and radii a and b, for th right and th lft nds, rspctivly, as shown in th Figur Figur 6.5. A truncatd Con. Assuming that th currnt is distributd uniformly throughout th cross-sction of th con, what is th rsistanc btwn th two nds? Solution: Considr a thin disk of radius r at a distanc x from th lft nd. From th figur shown on th right, w hav b r b a = x h or r = ( a b) x + b h Sinc rsistanc R is rlatd to rsistivity ρ by R = ρl/ A, whr l is th lngth of th conductor and A is th cross sction, th contribution to th rsistanc from th disk having a thicknss dy is ρ dx dr = = π π ρ dx + r [ b ( a b) x/ h] 11
13 Straightforward intgration thn yilds h ρ dx ρh R = = 0 π[ b+ ( a b) x/ h] π ab whr w hav usd du 1 = α + β α α + β ( u ) ( u ) Not that if b= a, Eq. (6..9) is rproducd Rsistanc of a Hollow Cylindr Considr a hollow cylindr of lngth L and innr radius a and outr radius b, as shown in Figur Th matrial has rsistivity ρ. Figur A hollow cylindr. (a) Suppos a potntial diffrnc is applid btwn th nds of th cylindr and producs a currnt flowing paralll to th axis. What is th rsistanc masurd? (b) If instad th potntial diffrnc is applid btwn th innr and outr surfacs so that currnt flows radially outward, what is th rsistanc masurd? Solution: (a) Whn a potntial diffrnc is applid btwn th nds of th cylindr, currnt flows paralll to th axis. In this cas, th cross-sctional ara is A= π ( b a ), and th rsistanc is givn by ρl ρl R = = A π b a ( ) 1
14 (b) Considr a diffrntial lmnt which is mad up of a thin cylindr of innr radius r and outr radius r + dr and lngth L. Its contribution to th rsistanc of th systm is givn by ρ dl ρ dr dr = = A π rl whr A= π rl is th ara normal to th dirction of currnt flow. Th total rsistanc of th systm bcoms b ρdr ρ b R = = ln a πrl πl a 6.6 Concptual Qustions 1. Two wirs A and B of circular cross-sction ar mad of th sam mtal and hav qual lngths, but th rsistanc of wir A is four tims gratr than that of wir B. Find th ratio of thir cross-sctional aras.. From th point of viw of atomic thory, xplain why th rsistanc of a matrial incrass as its tmpratur incrass. 3. Two conductors A and B of th sam lngth and radius ar connctd across th sam potntial diffrnc. Th rsistanc of conductor A is twic that of B. To which conductor is mor powr dlivrd? 6. Additional Problms 6..1 Currnt and Currnt Dnsity 9 A sphr of radius 10 mm that carris a charg of 8 nc = 8 10 C is whirld in a circl at th nd of an insulatd string. Th rotation frquncy is 100π rad/s. (a) What is th basic dfinition of currnt in trms of charg? (b) What avrag currnt dos this rotating charg rprsnt? (c) What is th avrag currnt dnsity ovr th ara travrsd by th sphr? 6.. Powr Loss and Ohm s Law A 1500 W radiant hatr is constructd to oprat at 115 V. 13
15 (a) What will b th currnt in th hatr? [Ans. ~10 A] (b) What is th rsistanc of th hating coil? [Ans. ~10 Ω] (c) How many kilocaloris ar gnratd in on hour by th hatr? (1 Calori = 4.18 J) 6..3 Rsistanc of a Con A coppr rsistor of rsistivity ρ is in th shap of a cylindr of radius b and lngth L 1 appndd to a truncatd right circular con of lngth L and nd radii b and a as shown in Figur Figur 6..1 (a) What is th rsistanc of th cylindrical portion of th rsistor? (b) What is th rsistanc of th ntir rsistor? (Hint: For th taprd portion, it is ncssary to writ down th incrmntal rsistanc dr of a small slic, dx, of th rsistor at an arbitrary position, x, and thn to sum th slics by intgration. If th tapr is small, on may assum that th currnt dnsity is uniform across any cross sction.) (c) Show that your answr rducs to th xpctd xprssion if a = b. (d) If L 1 = 100 mm, L = 50 mm, a = 0.5 mm, b = 1.0 mm, what is th rsistanc? 6..4 Currnt Dnsity and Drift Spd (a) A group of chargs, ach with charg q, movs with vlocity v. Th numbr of particls pr unit volum is n. What is th currnt dnsity J of ths chargs, in magnitud and dirction? Mak sur that your answr has units of A/m. (b) W want to calculat how long it taks an lctron to gt from a car battry to th startr motor aftr th ignition switch is turnd. Assum that th currnt flowing is115 A, and that th lctrons travl through coppr wir with cross-sctional ara 31. mm and lngth 85.5 c m. What is th currnt dnsity in th wir? Th numbr dnsity of th 8 3 conduction lctrons in coppr is /m. Givn this numbr dnsity and th currnt dnsity, what is th drift spd of th lctrons? How long dos it tak for an 14
16 6 lctron starting at th battry to rach th startr motor? [Ans: A/m, m/s,5.5 min.] 6..5 Currnt Sht A currnt sht, as th nam implis, is a plan containing currnts flowing in on dirction in that plan. On way to construct a sht of currnt is by running many paralll wirs in a plan, say th yz -plan, as shown in Figur 6..(a). Each of ths wirs carris currnt I out of th pag, in th j ˆ dirction, with n wirs pr unit lngth in th z-dirction, as shown in Figur 6..(b). Thn th currnt pr unit lngth in th z dirction is ni. W will us th symbol K to signify currnt pr unit lngth, so that K = nl hr. Figur 6.. A currnt sht. Anothr way to construct a currnt sht is to tak a non-conducting sht of charg with fixd charg pr unit ara σ and mov it with som spd in th dirction you want currnt to flow. For xampl, in th sktch to th lft, w hav a sht of charg moving out of th pag with spd v. Th dirction of currnt flow is out of th pag. (a) Show that th magnitud of th currnt pr unit lngth in th z dirction, K, is givn by σ v. Chck that this quantity has th propr dimnsions of currnt pr lngth. This is in fact a vctor rlation, K (t) = σ v( t), sinc th sns of th currnt flow is in th sam dirction as th vlocity of th positiv chargs. (b) A blt transfrring charg to th high-potntial innr shll of a Van d Graaff acclrator at th rat of.83 mc/s. If th width of th blt carrying th charg is 50 cm and th blt travls at a spd of 30 m /s, what is th surfac charg dnsity on th blt? [Ans: 189 µc/m ] 6..6 Rsistanc and Rsistivity A wir with a rsistanc of 6.0 Ω is drawn out through a di so that its nw lngth is thr tims its original lngth. Find th rsistanc of th longr wir, assuming that th 15
17 rsistivity and dnsity of th matrial ar not changd during th drawing procss. [Ans: 54 Ω]. 6.. Powr, Currnt, and Voltag A 100-W light bulb is pluggd into a standard 10-V outlt. (a) How much dos it cost pr month (31 days) to lav th light turnd on? Assum lctricity costs 6 cnts pr kw h. (b) What is th rsistanc of th bulb? (c) What is th currnt in th bulb? [Ans: (a) $4.46; (b) 144 Ω; (c) A] Charg Accumulation at th Intrfac Figur 6..3 shows a thr-layr sandwich mad of two rsistiv matrials with rsistivitis ρ 1 and ρ. From lft to right, w hav a layr of matrial with rsistivity ρ 1 of width d /3, followd by a layr of matrial with rsistivity ρ, also of width d /3, followd by anothr layr of th first matrial with rsistivity ρ 1, again of width d /3. Figur 6..3 Charg accumulation at intrfac. Th cross-sctional ara of all of ths matrials is A. Th rsistiv sandwich is boundd on ithr sid by mtallic conductors (black rgions). Using a battry (not shown), w maintain a potntial diffrnc V across th ntir sandwich, btwn th mtallic conductors. Th lft sid of th sandwich is at th highr potntial (i.., th lctric filds point from lft to right). Thr ar four intrfacs btwn th various matrials and th conductors, which w labl a through d, as indicatd on th sktch. A stady currnt I flows through this sandwich from lft to right, corrsponding to a currnt dnsity J = I / A. (a) What ar th lctric filds E 1 and E in th two diffrnt dilctric matrials? To obtain ths filds, assum that th currnt dnsity is th sam in vry layr. Why must this b tru? [Ans: All filds point to th right, E 1 = ρ 1 I / A, E = ρ I / A; th currnt dnsitis must b th sam in a stady stat, othrwis thr would b a continuous buildup of charg at th intrfacs to unlimitd valus.] 16
18 (b) What is th total rsistanc R of this sandwich? Show that your xprssion rducs to th xpctd rsult if ρ 1 = ρ = ρ. [Ans: R= d( ρ1+ ρ) / 3 A; if ρ 1 = ρ = ρ, thn R = d ρ / A, as xpctd.] (c) As w mov from right to lft, what ar th changs in potntial across th thr V ρ / ρ ρ,v ρ / ρ + ρ, layrs, in trms of V and th rsistivitis? [Ans: 1 ( 1+ ) ( 1 ) V ρ /( ρ + ρ ), summing to a total potntial drop of V, as rquird]. 1 1 (d) What ar th chargs pr unit ara, σ a through σ d, at th intrfacs? Us Gauss's Law and assum that th lctric fild in th conducting caps is zro. [Ans: σ a = σd = 3 ε0vρ1/ d( ρ1+ ρ), σ b = σc = 3 ε0v( ρ ρ1) / d ( ρ1+ ρ).] () Considr th limit ρ ρ 1. What do your answrs abov rduc to in this limit? 1
6 Chapter. Current and Resistance
6 Chapter Current and Resistance 6.1 Electric Current... 6-2 6.1.1 Current Density... 6-2 6.2 Ohm s Law... 6-5 6.3 Summary... 6-8 6.4 Solved Problems... 6-9 6.4.1 Resistivity of a Cable... 6-9 6.4.2 Charge
More informationElectromagnetism Physics 15b
lctromagntism Physics 15b Lctur #8 lctric Currnts Purcll 4.1 4.3 Today s Goals Dfin lctric currnt I Rat of lctric charg flow Also dfin lctric currnt dnsity J Charg consrvation in a formula Ohm s Law vryon
More informationVoltage, Current, Power, Series Resistance, Parallel Resistance, and Diodes
Lctur 1. oltag, Currnt, Powr, Sris sistanc, Paralll sistanc, and Diods Whn you start to dal with lctronics thr ar thr main concpts to start with: Nam Symbol Unit oltag volt Currnt ampr Powr W watt oltag
More informationPHYS ,Fall 05, Term Exam #1, Oct., 12, 2005
PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What
More informationPH2200 Practice Final Exam Spring 2004
PH2200 Practic Final Exam Spring 2004 Instructions 1. Writ your nam and studnt idntification numbr on th answr sht. 2. This a two-hour xam. 3. Plas covr your answr sht at all tims. 4. This is a closd book
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationEAcos θ, where θ is the angle between the electric field and
8.4. Modl: Th lctric flux flows out of a closd surfac around a rgion of spac containing a nt positiv charg and into a closd surfac surrounding a nt ngativ charg. Visualiz: Plas rfr to Figur EX8.4. Lt A
More informationPart 7: Capacitance And Capacitors
Part 7: apacitanc And apacitors 7. Elctric harg And Elctric Filds onsidr a pair of flat, conducting plats, arrangd paralll to ach othr (as in figur 7.) and sparatd by an insulator, which may simply b air.
More information7.4 Potential Difference and Electric Potential
7.4 Potntial Diffrnc and Elctric Potntial In th prvious sction, you larnd how two paralll chargd surfacs produc a uniform lctric fild. From th dfinition of an lctric fild as a forc acting on a charg, it
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationDeepak Rajput
Q Prov: (a than an infinit point lattic is only capabl of showing,, 4, or 6-fold typ rotational symmtry; (b th Wiss zon law, i.. if [uvw] is a zon axis and (hkl is a fac in th zon, thn hu + kv + lw ; (c
More informationPHYS-333: Problem set #2 Solutions
PHYS-333: Problm st #2 Solutions Vrsion of March 5, 2016. 1. Visual binary 15 points): Ovr a priod of 10 yars, two stars sparatd by an angl of 1 arcsc ar obsrvd to mov through a full circl about a point
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More informationHigh Energy Physics. Lecture 5 The Passage of Particles through Matter
High Enrgy Physics Lctur 5 Th Passag of Particls through Mattr 1 Introduction In prvious lcturs w hav sn xampls of tracks lft by chargd particls in passing through mattr. Such tracks provid som of th most
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationCurrent and Resistance
7 Currnt and Rsistanc CHPTER OUTLNE 71 Elctric Currnt 7 Rsistanc 7 Modl for Elctrical Conduction 74 Rsistanc and Tmpratur 75 Suprconductors 76 Elctric Powr NSWERS TO QUESTONS Q71 ndividual vhicls cars,
More informationELECTROMAGNETIC INDUCTION CHAPTER - 38
. (a) CTOMAGNTIC INDUCTION CHAPT - 38 3 3.dl MT I M I T 3 (b) BI T MI T M I T (c) d / MI T M I T. at + bt + c s / t Volt (a) a t t Sc b t Volt c [] Wbr (b) d [a., b.4, c.6, t s] at + b. +.4. volt 3. (a)
More informationA 1 A 2. a) Find the wavelength of the radio waves. Since c = f, then = c/f = (3x10 8 m/s) / (30x10 6 Hz) = 10m.
1. Young s doubl-slit xprint undrlis th instrunt landing syst at ost airports and is usd to guid aircraft to saf landings whn th visibility is poor. Suppos that a pilot is trying to align hr plan with
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationLecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e
8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationIntroduction to the quantum theory of matter and Schrödinger s equation
Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics
More informationPhysics 312 First Pledged Problem Set
Physics 31 First Pldgd Problm St 1. Th ground stat of hydrogn is dscribd by th wavfunction whr a is th Bohr radius. (a) Comput th charg dnsity à (r) = 1 p ¼ µ 1 a 3 r=a ; ½ (r) = jã (r)j : and plot 4¼r
More information5 Chapter Capacitance and Dielectrics
5 Chaptr Capacitanc and Dilctrics 5.1 Introduction... 5-3 5. Calculation of Capacitanc... 5-4 Exampl 5.1: Paralll-Plat Capacitor... 5-4 Exampl 5.: Cylindrical Capacitor... 5-6 Exampl 5.3: Sphrical Capacitor...
More informationChapter 6: Polarization and Crystal Optics
Chaptr 6: Polarization and Crystal Optics * P6-1. Cascadd Wav Rtardrs. Show that two cascadd quartr-wav rtardrs with paralll fast axs ar quivalnt to a half-wav rtardr. What is th rsult if th fast axs ar
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationElectrical Energy and Capacitance
haptr 6 Elctrical Enrgy and apacitanc Quick Quizzs. (b). Th fild xrts a forc on th lctron, causing it to acclrat in th dirction opposit to that of th fild. In this procss, lctrical potntial nrgy is convrtd
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationLast time. Resistors. Circuits. Question. Quick Quiz. Quick Quiz. ( V c. Which bulb is brighter? A. A B. B. C. Both the same
Last tim Bgin circuits Rsistors Circuits Today Rsistor circuits Start rsistor-capacitor circuits Physical layout Schmatic layout Tu. Oct. 13, 2009 Physics 208 Lctur 12 1 Tu. Oct. 13, 2009 Physics 208 Lctur
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationWhy is a E&M nature of light not sufficient to explain experiments?
1 Th wird world of photons Why is a E&M natur of light not sufficint to xplain xprimnts? Do photons xist? Som quantum proprtis of photons 2 Black body radiation Stfan s law: Enrgy/ ara/ tim = Win s displacmnt
More informationPHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS
PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More information0WAVE PROPAGATION IN MATERIAL SPACE
0WAVE PROPAGATION IN MATERIAL SPACE All forms of EM nrgy shar thr fundamntal charactristics: 1) thy all tral at high locity 2) In traling, thy assum th proprtis of was 3) Thy radiat outward from a sourc
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationExam 2 Thursday (7:30-9pm) It will cover material through HW 7, but no material that was on the 1 st exam.
Exam 2 Thursday (7:30-9pm) It will covr matrial through HW 7, but no matrial that was on th 1 st xam. What happns if w bash atoms with lctrons? In atomic discharg lamps, lots of lctrons ar givn kintic
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationDerivation of Electron-Electron Interaction Terms in the Multi-Electron Hamiltonian
Drivation of Elctron-Elctron Intraction Trms in th Multi-Elctron Hamiltonian Erica Smith Octobr 1, 010 1 Introduction Th Hamiltonian for a multi-lctron atom with n lctrons is drivd by Itoh (1965) by accounting
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationA central nucleus. Protons have a positive charge Electrons have a negative charge
Atomic Structur Lss than ninty yars ago scintists blivd that atoms wr tiny solid sphrs lik minut snookr balls. Sinc thn it has bn discovrd that atoms ar not compltly solid but hav innr and outr parts.
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationMA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
More informationVSMN30 FINITA ELEMENTMETODEN - DUGGA
VSMN3 FINITA ELEMENTMETODEN - DUGGA 1-11-6 kl. 8.-1. Maximum points: 4, Rquird points to pass: Assistanc: CALFEM manual and calculator Problm 1 ( 8p ) 8 7 6 5 y 4 1. m x 1 3 1. m Th isotropic two-dimnsional
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationDefinition1: The ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.
Dirctivity or Dirctiv Gain. 1 Dfinition1: Dirctivity Th ratio of th radiation intnsity in a givn dirction from th antnna to th radiation intnsity avragd ovr all dirctions. Dfinition2: Th avg U is obtaind
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More informationChapter 8: Electron Configurations and Periodicity
Elctron Spin & th Pauli Exclusion Principl Chaptr 8: Elctron Configurations and Priodicity 3 quantum numbrs (n, l, ml) dfin th nrgy, siz, shap, and spatial orintation of ach atomic orbital. To xplain how
More informationSystem variables. Basic Modeling Concepts. Basic elements single and. Power = effort x flow. Power = F x v. Power = V x i. Power = T x w.
Basic Modling Concpts Basic lmnts singl and multiport t dvics Systm variabls v m F V i Powr F x v T w Powr T x w Powr V x i P Q Powr P x Q Powr ort x low Eort & low ar powr variabls Eorts t... Flows...
More informationsurface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationElectrochemistry L E O
Rmmbr from CHM151 A rdox raction in on in which lctrons ar transfrrd lctrochmistry L O Rduction os lctrons xidation G R ain lctrons duction W can dtrmin which lmnt is oxidizd or rducd by assigning oxidation
More informationMaxwellian Collisions
Maxwllian Collisions Maxwll ralizd arly on that th particular typ of collision in which th cross-sction varis at Q rs 1/g offrs drastic siplifications. Intrstingly, this bhavior is physically corrct for
More informationCoupled Pendulums. Two normal modes.
Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationStructure of the Atom. Thomson s Atomic Model. Knowledge of atoms in Experiments of Geiger and Marsden 2. Experiments of Geiger and Marsden
CHAPTER 4 Structur of th Atom 4.1 Th Atomic Modls of Thomson and Ruthrford 4. Ruthrford Scattring 4.3 Th Classic Atomic Modl 4.4 Th Bohr Modl of th Hydrogn Atom 4.5 Succsss & Failurs of th Bohr Modl 4.6
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationDIELECTRIC AND MAGNETIC PROPERTIES OF MATERIALS
DILCTRIC AD MAGTIC PROPRTIS OF MATRIALS Dilctric Proprtis: Dilctric matrial Dilctric constant Polarization of dilctric matrials, Typs of Polarization (Polarizability). quation of intrnal filds in liquid
More informationElectrochemical Energy Systems Spring 2014 MIT, M. Z. Bazant. Midterm Exam
10.66 Elctrochmical Enrgy Systms Spring 014 MIT, M. Z. Bazant Midtrm Exam Instructions. This is a tak-hom, opn-book xam du in Lctur. Lat xams will not b accptd. You may consult any books, handouts, or
More informationSCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER. J. C. Sprott
SCALING OF SYNCHROTRON RADIATION WITH MULTIPOLE ORDER J. C. Sprott PLP 821 Novmbr 1979 Plasma Studis Univrsity of Wisconsin Ths PLP Rports ar informal and prliminary and as such may contain rrors not yt
More informationPreliminary Fundamentals
1.0 Introduction Prliminary Fundamntals In all of our prvious work, w assumd a vry simpl modl of th lctromagntic torqu T (or powr) that is rquird in th swing quation to obtain th acclrating torqu. This
More informationSeptember 23, Honors Chem Atomic structure.notebook. Atomic Structure
Atomic Structur Topics covrd Atomic structur Subatomic particls Atomic numbr Mass numbr Charg Cations Anions Isotops Avrag atomic mass Practic qustions atomic structur Sp 27 8:16 PM 1 Powr Standards/ Larning
More informationTitle: Vibrational structure of electronic transition
Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationDivision of Mechanics Lund University MULTIBODY DYNAMICS. Examination Name (write in block letters):.
Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions
More informationECE507 - Plasma Physics and Applications
ECE507 - Plasma Physics and Applications Lctur 7 Prof. Jorg Rocca and Dr. Frnando Tomasl Dpartmnt of Elctrical and Computr Enginring Collisional and radiativ procsss All particls in a plasma intract with
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationCollisions between electrons and ions
DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationUniversity of Illinois at Chicago Department of Physics. Thermodynamics & Statistical Mechanics Qualifying Examination
Univrsity of Illinois at Chicago Dpartmnt of hysics hrmodynamics & tatistical Mchanics Qualifying Eamination January 9, 009 9.00 am 1:00 pm Full crdit can b achivd from compltly corrct answrs to 4 qustions.
More informationGive the letter that represents an atom (6) (b) Atoms of A and D combine to form a compound containing covalent bonds.
1 Th diagram shows th lctronic configurations of six diffrnt atoms. A B C D E F (a) You may us th Priodic Tabl on pag 2 to hlp you answr this qustion. Answr ach part by writing on of th lttrs A, B, C,
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationExtraction of Doping Density Distributions from C-V Curves
Extraction of Doping Dnsity Distributions from C-V Curvs Hartmut F.-W. Sadrozinski SCIPP, Univ. California Santa Cruz, Santa Cruz, CA 9564 USA 1. Connction btwn C, N, V Start with Poisson quation d V =
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME Introduction to Finit Elmnt Analysis Chaptr 5 Two-Dimnsional Formulation Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More information