GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS ` E MISN CHARGE DISTRIBUTIONS by Peter Signell, Michigan State University
|
|
- Richard Lloyd
- 7 years ago
- Views:
Transcription
1 MISN GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction a. Oveview b. Usefulness Cylindical Symmety: Line Chage a. Appoximating a Real Line by an Infinite One b. The Gaussian Suface c. The Electic Field A P ` E 3. Othe Cylindical Distibutions a. Electic Field of a Cylindical Suface b. Linea vs. Suface Chage Density c. The Coaxial Cable d. Electic Field of the Coaxial Cable A Single Sheet of Chage a. Appoximation: An Infinite Sheet b. The Gaussian Suface c. The Electic Field Two Paallel Sheets of Chage a. Unequal Suface Chage Densities b. Equal Suface Chage Densities Acknowledgments Glossay Poject PHYSNETPhysics Bldg. Michigan State UnivesityEast Lansing, MI 1
2 ID Sheet: MISN Title: Gauss s Law Applied to Cylindical and Plana Chage Distibutions Autho: P. Signell, Dept. of Physics, Mich. State Univ Vesion: 2/28/2000 Evaluation: Stage 0 Length: 1 h; 24 pages Input Skills: 1. Vocabulay: cylindical symmety, plana symmety (MISN-0-153); Gaussian suface, volume chage density (MISN-0-132). 2. State Gauss s law and apply it in cases of spheical symmety (MISN-0-132). Output Skills (Knowledge): K1. Vocabulay: coaxial cable, cylinde of chage, line of chage, sheet of chage, linea chage density. K2. Justify the Gaussian-Suface shapes that ae appopiate fo cylindical and plana chage distibutions. K3. State Gauss s Law in equation fom and define each symbol. Fo cylindical and plana chage distibutions, define needed paametes and then, justifying each step as you go, solve Gauss s Law fo the symbolic electic field at a space-point. Output Skills (Poblem Solving): S1. Given a specific chage distibution with cylindical o plana symmety, use Gauss s law to detemine the electic field poduced by the chage distibution. Post-Options: 1. Electic Fields and Potentials Acoss Chage Layes and In Capacitos (MISN-0-134). 2. Electostatic Capacitance (MISN-0-135). THIS IS A DEVELOPMENTAL-STAGE PUBLICATION OF PROJECT PHYSNET The goal of ou poject is to assist a netwok of educatos and scientists in tansfeing physics fom one peson to anothe. We suppot manuscipt pocessing and distibution, along with communication and infomation systems. We also wok with employes to identify basic scientific skills as well as physics topics that ae needed in science and technology. A numbe of ou publications ae aimed at assisting uses in acquiing such skills. Ou publications ae designed: (i) to be updated quickly in esponse to field tests and new scientific developments; (ii) to be used in both classoom and pofessional settings; (iii) to show the peequisite dependencies existing among the vaious chunks of physics knowledge and skill, as a guide both to mental oganization and to use of the mateials; and (iv) to be adapted quickly to specific use needs anging fom single-skill instuction to complete custom textbooks. New authos, eviewes and field testes ae welcome. PROJECT STAFF Andew Schnepp Eugene Kales Pete Signell Webmaste Gaphics Poject Diecto ADVISORY COMMITTEE D. Alan Bomley Yale Univesity E. Leonad Jossem The Ohio State Univesity A. A. Stassenbug S. U. N. Y., Stony Book Views expessed in a module ae those of the module autho(s) and ae not necessaily those of othe poject paticipants. c 2001, Pete Signell fo Poject PHYSNET, Physics-Astonomy Bldg., Mich. State Univ., E. Lansing, MI 48824; (517) Fo ou libeal use policies see: 3 4
3 MISN GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction 1a. Oveview. In this module Gauss s law is used to find the electic field in the neighbohood of chage distibutions that have cylindical and plana symmety. Fo each of these two symmeties, useful Gaussian sufaces ae easily constucted. Once an appopiate Gaussian suface is constucted, the electic field is easily found fom Gauss s law: 1 E ˆn ds = 4πk e q S (1) whee q S is the net chage enclosed by the Gaussian suface S and k e is the electostatic foce constant. 1b. Usefulness. It is vey useful to know the electic field in the neighbohood of cylindical and plana chage distibutions, fo these geometies ae the ones used in coaxial cables and capacitos. Knowing the electic fields helps one detemine how these devices will eact in electonic cicuits. In addition, the same geneal ideas ae used in detemining the magnetic fields poduced in solenoids, tansfomes, coaxial cables, chokes, and tansmission lines. 2. Cylindical Symmety: Line Chage 2a. Appoximating a Real Line by an Infinite One. When dealing with a line of chage, we will teat it as though its ends had been extended to infinity. This appoximation makes the esulting electic field especially simple and easy to solve fo. The solutions we get fo the infinitely long line will be applicable to the finite-line case fo electic field points that ae much close to the middle pat of the line than to its ends. Fo pactical applications the infinite line is almost always a good appoximation to the actual finite line. 1 See Gauss s Law and Spheically Symmetic Chage Distibutions (MISN-0-132) fo an intoduction to Gauss s law and the ules fo using it. MISN Gaussian suface line of chage Figue 1. A cylindical Gaussian suface is used to apply Gauss s law to a line of chage. 2b. The Gaussian Suface. Fo an infinitely long line with unifom linea chage density 2 along it, the pefeed Gaussian suface is cylindical (see Fig. 1). This follows fom taking the two ules fo constucting Gaussian sufaces and combining them with knowledge of the electic field s diections and equi-magnitude sufaces. 3 The axis of the cylindical suface is along the line of chage, while the suface s adius is that of the point at which you wish to know the electic field. The length of the cylindical suface is immateial. 2c. The Electic Field. Applying Gauss s law, Eq. (1), to the case of a staight line of chage with unifom linea chage density (chage pe unit length) λ, we will show that the magnitude of the electic field at a distance fom the line is: E = 2k e λ. (2) Poof: If the length of the cylindical Gaussian suface is L, then the chage enclosed by the suface is: q S = λ L. (3) The component of the electic field nomal to eithe flat end of the closed cylindical suface is zeo, but the component nomal to the cylindical 2 The tem linea chage density means the chage is being descibed as a cetain amount of chage pe unit length along the wie. This is in contast to volume chage density whee the chage is descibed as a cetain amount of chage pe unit volume within the wie. Fo unifom coss-sectional distibutions, the linea chage density equals the coss-sectional aea times the volume chage density. 3 Fo the two ules fo constucting Gaussian sufaces, see Ref. 1. Fo deivation of the electic field diections and equi-magnitude sufaces see Electic Fields fom Symmetic Chage Distibutions (MISN-0-153). 5 6
4 MISN pat of the suface is just the field itself: E ˆn ds = E ˆn ds E ˆn ds = E cyl. ends cyl. ds 0 = E(2πL). (4) Using Gauss s law, Eq. (1), to combine Eqs. (3) and (4), we obtain the solution, Eq. (2). Of couse in a eal poblem ou solution would be valid only in the egion whee the distance to the line of chage is much smalle than the distance to the line s neaest end. As an amusing aside, notice that Eq. (2) says that the sound of a long line of taffic will only die off as 1 athe than the 2 one obtains fo a point souce. 3. Othe Cylindical Distibutions 3a. Electic Field of a Cylindical Suface. A cylindical suface with finite adius, constant suface chage density, and infinite extent, has an electic field whose pefeed Gaussian sufaces ae identical to those fo an infinite chaged line (see Fig. 2). This is because both the line and the cylindical suface have the same geometical symmety and hence the same electic field diections and equi-magnitude sufaces. 4 Fo a chaged suface of adius R and suface chage density σ, the amount of chage 4 See Electic Field fom Symmetic Chage Distibutions, (MISN-0-153), the section on infinitely long cylindical chage distibutions. R Gaussian suface cylindical suface of chage Figue 2. The Gaussian suface fo a cylindical suface chage distibution (on a cylindical suface of adius R). MISN enclosed by a Gaussian suface of adius and length L is: q S = 2πR L σ fo > R = 0 fo < R Exactly as in the case of the line of chage, the integal of the nomal component of the electic field ove the Gaussian suface is: E ˆn ds = (2πL)E. (6) Then using Eqs. (5) and (6) in Gauss s law, Eq. (1), we find: σr E = 4πk e fo > R = 0 fo < R 3b. Linea vs. Suface Chage Density. We may descibe the chage distibution on a cylindical suface as eithe a suface chage density o a linea chage density. The suface chage density σ is the chage pe unit aea on the cylindical suface: σ = q (8) 2πl whee 2πl is the suface aea of a cylinde of adius and length l. The linea chage density λ is the (total) chage pe unit length along the cylindical suface: λ = q = 2πσ. (9) l Show that if q = C, = 1.0 cm, and l = 1.0 m, then σ = C/m 2 and λ = C/m. 3c. The Coaxial Cable. A coaxial cable, such as that used to tanspot TV signals o the signal fom a pickup to a steeo amplifie, consists of two metallic conductos with cylindical symmety, shaing a common cylinde axis and sepaated by some kind of insulato. The cente cylinde is usually a solid coppe wie while the oute one is usually a sheath of baided wie (see Fig. 3). This constuction makes the cable mechanically flexible. As fa as the electical popeties ae concened, thee might as well be two concentic cylindical sufaces 5 as shown in Fig. 4. The magnitude of the linea chage density on the two cylindes is the same, so the 5 Electostatic chages eside on the sufaces of metallic conductos: see Electostatic Popeties of Conductos (MISN-0-136). (5) (7) 7 8
5 MISN MISN dielectic (usually white, flexible) cente wie (solid) baided wie sheath plastic skin Figue 3. A coss-sectional view of a typical coaxial cable. I II III Figue 4. A cosssectional view of the chage sufaces in a coaxial cable. The adius of the inne cylinde is exaggeated fo the pupose of illustation. magnitude of the suface chage density is highe on the inne cylinde. Fo odinay uses the chages on the two cylindes ae of opposite sign. 3d. Electic Field of the Coaxial Cable. Gauss s law shows that the electic field of a chaged coaxial cable is zeo except between the conducting cylindical sufaces, whee it is equal to the field poduced by the inne cylinde. Applying Gauss s law to the coaxial cable s vaious egions, as shown in Fig. 4, the electic field is eadily found to be: E I = 0 λ E II = 2k e ˆ E III = 0 Help: [S-1] (10) In egion II of Fig. 4, λ is negative, so that paticula electic field is diected adially inwad. a) b) ` E n^ n^ C A B n^ P ` E Figue 5. (a) A coss-sectional view of a Gaussian suface (dashed lines) fo an infinite plane of chage; and (b) a theedimensional view of the Gaussian suface. 4. A Single Sheet of Chage 4a. Appoximation: An Infinite Sheet. We will estict ouselves to the case of a unifom plana chage distibution of infinite extent: in othe wods, a flat sheet with a unifom suface chage density that extends to infinity. These estictions make the esulting electic field especially simple and easy to detemine. The solutions we get will be valid fo any application in which the sheet of chage can be appoximated by an infinite sheet with the same suface chage density. This appoximation will be a good one when the distances fom elevant electic field points to the edges of the physical sheet ae all much lage than the distance to the neaest point on the sheet of chage. Thus the edges will look an almost infinite distance away (in compaison). This will be the case fo impotant chage-stoing components in electonic cicuits. 4b. The Gaussian Suface. Fo a unifom plana chage distibution of infinite extent, all pats of the pefeed Gaussian suface can be shown to be eithe paallel o pependicula to the plane of the chage. This equiement would be satisfied, fo example, by a box-like suface that is cut by the plane into two equal boxes (see Fig. 5). The paallel o pependicula equiement follows fom taking the two ules fo constucting Gaussian sufaces and combining them with ou knowledge of the electic field s diections and equi-magnitude sufaces. Any suface that satisfies the paallel o pependicula equiement is acceptable, C A B P ` E 9 10
6 MISN but ectangula and cylindical boxes ae the easiest to use in computing the suface aeas and volumes that ente into Gauss s law. 4c. The Electic Field. Applying Gauss s law to a flat infinite sheet with unifom suface chage density σ, we find that the magnitude of the electic field is eveywhee the same: E = 2πk e σ. (11) The diection of the field is nomal to the sheet of chage, diected away fom the sheet fo a positive chage density and towad the sheet fo a negative chage density. If the aea of one end of the box-like Gaussian suface is A, then the chage enclosed by the suface is: q S = σ A. (12) The component of the electic field nomal to the side of the Gaussian suface is zeo on the fou sides that cut though the plane and equal to the electic field on the othe two sides: E ˆn ds = 2 E A. (13) Using Gauss s law to combine Eqs. (12) and (13) we obtain Eq. (11), the solution. Of couse in a eal poblem the constancy of the electic field is esticted to egions whee the distance to the sheet of chage is much smalle than the distance to the sheet s neaest edge. 5. Two Paallel Sheets of Chage 5a. Unequal Suface Chage Densities. Gauss s law can be easily applied to the case of two infinite paallel sheets having unifom suface chage densities σ and σ, espectively. To obtain the electic field at some paticula point, apply Gauss s law to each of the sheets sepaately and then add the fields fom the two sheets vectoially. Note that the two Gaussian sufaces have one side in common. You should obtain the answes: Help: [S-3] E = 2πk e (σ σ ), (outside the planes). (14) E = 2πk e (σ σ ), (between the planes). (15) Note that these equations become paticulaly simple when σ and σ ae equal. Note also that this poblem has no symmety plane so a single pefeed Gaussian suface could not be dawn: the two ules fo constucting such a suface could not be satisfied with only ou usual pio knowledge of the field. MISN b. Equal Suface Chage Densities. Fo two infinite paallel sheets of chage with identical suface chage densities, σ, we can apply Gauss s law using a single Gaussian suface. Thee is a plane of symmety that is paallel to the two sheets and half way between them, so the Gaussian suface must be symmetical with espect to that plane of symmety. In pactical tems, the symmety plane must cut the boxlike suface into two identical box-like sufaces. Note how this symmety of the Gaussian suface with espect to the poblem s symmety plane ensues that the two ules fo constucting the pefeed suface can be satisfied. Help: [S-2] If one end of the suface has aea A, the chage enclosed by the suface is: q S = 2 σ A. (16) Then Gauss s law poduces: and E = 4πk e σ, (outside the sheets), (17) E = 0, (between the sheets). These two equations agee with Eqs. (14) and (15). Acknowledgments I would like to thank Pofesso J. Linnemann fo a valuable suggestion. Pepaation of this module was suppoted in pat by the National Science Foundation, Division of Science Education Development and Reseach, though Gant #SED to Michigan State Univesity. Glossay coaxial cable: an electical cable consisting of two metallic concentic cylindes sepaated by some kind of insulato. The electic field is zeo both inside the inne cylinde and outside the oute cylinde. cylinde of chage: a chage distibution with cylindical symmety. Fo a cylinde of chage with constant density extending to infinity, the associated electic field falls off as (1/) outside the suface ( is the adius fom the axis of the cylinde). line of chage: a chage distibution along a staight line. This is a special case of cylinde of chage
7 MISN linea chage density: the chage pe unit length along a line. sheet of chage: a chage distibution in a plane. Fo a plane of chage with constant density extending to infinity in all diections, the associated electic field is eveywhee constant and nomal to the plane. MISN PROBLEM SUPPLEMENT Note: Poblems 7 and 8 also occu in this module s Model Exam. PS-1 1. Two paallel lines of chage, shown coming out of the page in the sketch, ae a distance 2.0d apat. If both ae positively chaged, find the electic field as a function of y at points along the pependicula bisecto of the line connecting the two [find E(y) fo x = 0]. y d d x 2. An infinitely long cylinde of chage has a adius R and a volume chage density ρ. Find the electic field in the egions < R and > R and show that both lead to the same esult at = R. Help: [S-9] 3. Find the electic field inside and outside a hollow cylinde of chage whose adius is R, and whose linea chage density is λ. R R 13 14
8 MISN PS-2 MISN PS-3 4. A cylinde of chage has volume chage density ρ and adius R 1. Outside it and concentic with it is a cylindical suface of chage with adius R 2. The chage on this oute suface has a linea chage density λ. Find the electic field in each of the thee egions defined by the cylinde sufaces at R 1 and R 2. Show that the esult in Poblem 3 can be obtained fom the esults in this Poblem by letting R 1 = 0, R 2 = R. 5. Find the electic field in the fou egions defined by the thee infinite planes (sheets) of chage in the sketch. y^ I II III IV x^ s - s s R 1 R 2 7. Use Gauss s law to find the electic field outside and inside two lage paallel plates with equal suface chage densities σ on the plates and with volume chage density ρ between the plates. The distance between the plates is d. Help: [S-11] 8. Use Gauss s law to find the electic field in each of the thee egions defined by two coaxial cylindical sufaces, each with linea chage density λ, and with a unifom volume chage density ρ inside the inne cylindical suface. The adii of the two cylindical sufaces ae R 1 and R 2 (see diagam below). R 2 R 1 volume chage x^ 6. An infinite slab of chage of thickness d has a unifom chage density ρ, as shown below in coss section in the sketch. Use Gauss s law to find the electic field inside and outside the slab. Help: [S-4] y d x 15 16
9 MISN PS-4 MISN PS-5 Bief Answes: E I = 2πk e σˆx ; Help: [S-8] 1. E x = 0 λy E y = 4k e (d 2 y 2 ) Help: [S-10] 2. E() = 2πke ρ ˆ fo < R Help: [S-6] ρr E() 2 = 2πk e ˆ fo > R At the suface: E(R) = 2πke ρr ˆ fo = R 3. Region 1 ( < R 1 ): E() = 2πke ρ ˆ Help: [S-7] Region 2 (R 1 < < R 2 ): E() = 2πke ρr 2 1 Region 3 (R 2 < ): E() ρπr = 1 2 λ 2ke ˆ ( ) ρr 2 1 = 0 ; E = 0 (fo < R) ) 4. lim R 1 0 2πk e lim R 1 0 ( ρr1 2 λ 2πk e 2k e ˆ λ = 2k e ; E() λ = 2k e ˆ (fo > R) 5. In the figue below, note that each ow of fou aows shows the field that would be poduced by just one plane of chage alone (see the annotations down the ight side of the figue). E II = 2πk e σˆx ; EIII = 2πk e σˆx ; EIV = 2πk e σˆx. 6. Inside : E = 4πke ρy ŷ ; Outside : E = 2πke ρd ŷ 7. Outside: Eight = 2πk e (2σ dρ) ˆx E left = 2πk e (2σ dρ) ˆx Inside: E = 4πke ρ x ˆx, whee x is measued fom the symmety plane between the plates. 8. Inside egion: E() = 2πke ρ ˆ Between egion: E() = 2ke πr 2 1ρ λ Outside egion: E() = 2ke πr 2 1ρ 2λ ˆ ˆ I II III IV field due to 1 field due to 2 field due to
10 MISN AS-1 MISN AS-2 SPECIAL ASSISTANCE SUPPLEMENT S-3 (fom TX-5a) Multiply all shown values by 2πk e : S-1 (fom TX-3d) S I EI ˆn ds = 4πk e q SI = 0 E I ˆn = 0 on cyl. suf. E I = 0 S II EII ˆn ds = 4πk e q SII = 4πk e λl E II ˆn(2πl) = 4πk e λl E II = 2k e λ ˆ S III EIII ˆn ds = 4πk e q SIII = 0 E III ˆn = 0 on cyl. suf. E III = 0 S-2 (fom TX-5b) S E ˆn ds = 4πk e q S = 4πk e 2σA on ends: E ˆn(2A) = 4πke 2σA outside the planes: E = 4πke σ ˆn S 1 E ˆn ds = 4πke q S = 0 on ends: E ˆn(2A) = 0 between the planes: E = 0 Region I Region II Region III n^ Cylindical Gaussian Suface n^ ' ' ' ' S-4 (fom PS-Poblem 6) ' - ' ' Since the slab has plana symmety, the field diection is eveywhee nomal to the slab and the equi-magnitude sufaces ae paallel to the slab faces (see MISN-0-153). When you constuct Gaussian sufaces, take advantage of the eflection symmety about the x-z plane by choosing the two faces that ae paallel to the slab to be equidistant fom the slab as well. That way you can wite the suface integal of E as: E ds = 2 E A S since the field is unifom ove the two Gaussian suface aeas, and the field must be the same on each suface by symmety. S-5 (fom [S-10]) Daw a sketch of the situation and on it mak the given quantities. Then figue out and mak on the sketch the θ and we use hee: E y (total) = E y (fom #1) E y (fom #2) E y (fom #1) = E(fom #1) cos θ E y (fom #2) = E(fom #2) cos θ E(fom #1) = E(fom #2) = 2k e λ/() whee = (d 2 y 2 ) 1/2, λ is the chage pe unit length along each wie, and cos θ = y/. E y (total) = 4k e λy/( 2 ) = 4k e λy/(d 2 y 2 ) 19 20
11 MISN AS-3 MISN ME-1 S-6 (fom PS-Poblem 2) Chage volume density is chage pe unit volume and in the MKS system is measued in C/m 3. Then: total chage = chage volume density volume, whee all thee quantities efe to inside the suface. S-7 (fom PS-Poblem 4) Completely solve Poblems 1 and 2 fist. Linea chage density is chage pe unit length and in the MKS system is measued in C/m. It is the amount of chage pe unit length down the entie cylindical suface. S-8 (fom PS-Poblem 5) Use Gauss s Law sepaately on each plane (as though the othe did not exist) to get: EI (due to #1), EI (due to #2), and E I (due to #3). Then add them to get E in egion I, hee labeled E I. S-9 (fom PS-poblem 2) The concept that has caused students touble in the past (in Poblem 2) is diectly and clealy handled in this module s text. Read and undestand it thee. S-10 (fom PS-poblem 1) Fist, ty you vey best to solve this poblem without Special Assistance. Go back and wok though the text again, this time paying special attention to sections elevant to this poblem. Remembe that you will not have the Special Assistance available at exam time so you need to lean how to wok without it. If you ty and ty and tuly fail, then ty the Special Assistance in [S-5]. S-11 (fom PS-poblem 7) Calculating the enclosed chage involves techniques you used in the two pevious poblems. MODEL EXAM 1. See Output Skill K1 in this module s ID Sheet. 2. Use Gauss s law to find the electic field outside and inside two lage paallel plates with equal suface chage densities σ on the plates and with volume chage density ρ between the plates. The distance between the plates is d. 3. Use Gauss s law to find the electic field in each of the thee egions defined by two coaxial cylindical sufaces, each with linea chage density λ, and with a unifom volume chage density ρ inside the inne cylindical suface. The adii of the two cylindical sufaces ae R 1 and R 2 (see diagam below). R 2 Bief Answes: R 1 1. See this module s text. volume chage 2. See this module s Poblem Supplement, Poblem 7. x^ 3. See this module s Poblem Supplement, Poblem
12 23 24
SELF-INDUCTANCE AND INDUCTORS
MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................
More informationGauss Law. Physics 231 Lecture 2-1
Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C
Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationCarter-Penrose diagrams and black holes
Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationVoltage ( = Electric Potential )
V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationVoltage ( = Electric Potential )
V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationA r. (Can you see that this just gives the formula we had above?)
24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.
More informationConcept and Experiences on using a Wiki-based System for Software-related Seminar Papers
Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationExperiment 6: Centripetal Force
Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationCharges, Coulomb s Law, and Electric Fields
Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationFinancing Terms in the EOQ Model
Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad
More informationLesson 8 Ampère s Law and Differential Operators
Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic
More informationModel Question Paper Mathematics Class XII
Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat
More informationWeek 3-4: Permutations and Combinations
Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationPAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary
PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationMoment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationElectrostatic properties of conductors and dielectrics
Unit Electostatic popeties of conductos and dielectics. Intoduction. Dielectic beaking. onducto in electostatic equilibium..3 Gound connection.4 Phenomena of electostatic influence. Electostatic shields.5
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationFunctions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem
Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationVISCOSITY OF BIO-DIESEL FUELS
VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationTECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications
JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationCRRC-1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer
CRRC- Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationStructure and evolution of circumstellar disks during the early phase of accretion from a parent cloud
Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The
More information7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what
More informationDo Vibrations Make Sound?
Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationThank you for participating in Teach It First!
Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident
More informationValuation of Floating Rate Bonds 1
Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationCollege of Engineering Bachelor of Computer Science
2 0 0 7 w w w. c n u a s. e d u College of Engineeing Bachelo of Compute Science This bochue Details the BACHELOR OF COMPUTER SCIENCE PROGRAM available though CNU s College of Engineeing. Fo ou most up-to-date
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More information(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m?
Em I Solutions PHY049 Summe 0 (Ch..5). Two smll, positively chged sphees hve combined chge of 50 μc. If ech sphee is epelled fom the othe by n electosttic foce of N when the sphees e.0 m pt, wht is the
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More informationSpirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project
Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationProblems of the 2 nd and 9 th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)
Poblems of the nd and 9 th Intenational Physics Olympiads (Budapest Hungay 968 and 976) Péte Vankó Institute of Physics Budapest Univesity of Technology and Economics Budapest Hungay Abstact Afte a shot
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability
More informationRisk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More information883 Brochure A5 GENE ss vernis.indd 1-2
ess x a eu / u e a. p o.eu c e / :/ http EURAXESS Reseaches in Motion is the gateway to attactive eseach caees in Euope and to a pool of wold-class eseach talent. By suppoting the mobility of eseaches,
More informationThings to Remember. r Complete all of the sections on the Retirement Benefit Options form that apply to your request.
Retiement Benefit 1 Things to Remembe Complete all of the sections on the Retiement Benefit fom that apply to you equest. If this is an initial equest, and not a change in a cuent distibution, emembe to
More information