PARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS
|
|
- Asher Goodwin
- 6 years ago
- Views:
Transcription
1 VISUAL PHYSICS ONLINE MODULE 6 ELECTROMAGNETISM PARTICLE MOTION IN UNIFORM GRAVITATIONAL and ELECTRIC FIELDS A fram of rfrnc Obsrvr Origin O(,, ) Cartsian coordinat as (X, Y, Z) Unit vctors iˆˆj k ˆ Scif th units Dfin zro oint for otntial nrg and lctric otntial Not: th magnitud of a vctor is alwas a ositiv numbr. Nwton s nd Law a 1 m F 1
2 Elctric forc FE q E Motion of a articl with a uniform acclration v u a t s ut at 1 v u a s s v t v u v Motion th XY lan with uniform acclration Al th quation for uniform acclration to th X Motion and Y Motion saratl a a iˆ a ˆj v v iˆ v ˆj s s iˆ s ˆj v v v v tan Enrgis v v Kintic nrg K 1 mv Elctric otntial nrg and lctric otntial (uniform lctric fild) Work E E ˆj U q E s V E s U q V W F s U q E s E W K q V
3 Total nrg E K U Consrvation of nrg K U Phsics is a bautiful subjct. Th sam laws, rincils and quations can oftn b alid to dscrib and rdict th bhaviour of a wid rang of hsical hnomna. As an aml, w will amin th motion of a articl in a uniform gravitational fild and th motion of chargd articls in uniform lctric filds and find that th actl sam laws and quations can b alid to both situations. 3
4 Motion in a uniform gravitational fild Nar th surfac of th Earth, to a good aroimation th gravitational forc F G acting on a articl of mass m is constant (uniform). W will invstigat th motion of a articl launchd from th Origin with an initial vlocit u in th uniform gravitational fild nar th surfac. W can rdict th vlocit v, dislacmnt s and nrgis K, U, E of th articl at an futur tim t. Fig. 1. Fram of rfrnc and forc acting on a articl in a uniform gravitational fild. Using Nwton s nd Law, th acclration of th articl is constant, bcaus th forc acting on th articl is constant FG a g ˆj a a g m 4
5 Not: Th acclration of th articl is indndnt of th articl s mass. Th motion of th articl is dscribd b th quations of uniform acclratd motion which ar alid to th motion in th X and Y dirctions a a g t s u u cos s u usin initial conditions v u v u a t Grah v vs t is a straight lin s v t s u t a t 1 Grah s vs t is a arabola v u a s u v v s v t avrag vlocit v v v v s s s s v tan v s tan s K mv mv mv kintic nrg U m g s s U gravitational otntial nrg E K U total nrg G work don b gravitational forc W F ds m g s 5
6 W K mv mu mv mu U W m g s K U consrvation of nrg 1 Prdict Obsrv Elain Ercis A articl of mass 1 kg is launchd with an initial vlocit of 1 m.s -1 at an angl of 75 o w.r.t. th horizontal. Calculat th tim, vlocit, osition, kintic nrg, otntial nrg and total nrg of th articl at th Origin, whn it rachs its maimum hight and whn it rturns to its launch hight. Prdict th sha of th grahs for: Trajctor (s vs s ) Tim grahs a, a v, v s, s K, U, E Mark on th grahs, whn th articl rachs its maimum hight and rturns to its launch hight. 6
7 Solution Us th grahs shown blow to chck our calculations and rdictd grahs. 7
8 8
9 9
10 Motion of chargd articls in uniform lctric filds How do lctric chargs mov in a uniform lctric fild? To gt startd in answring this qustion, w will invstigat th motion of an lctron and a roton in a uniform lctric fild. Th lctric fild btwn two oositl chargd aralll lats is aroimatl a uniform fild. As an aml, th otntial diffrnc btwn th lats is st at 1 V and th lat saration is mm. An lctron and a roton ar rlasd from rst at oints mid-wa btwn th two lats. Th zro osition for th otntial nrgis of th two chargs and th otntial is at th mid-oint btwn th lats whr th two chargs ar rlasd. Th hsical situation is shown in figur. Fig.. Motion of chargd articls in a uniform lctric fild. 1
11 Prdict Obsrv Elain Ercis Prdict th following aftr ach chargd articl has movd a distanc of 5. mm from thir initial rlas ositions. 1. Th location of ach charg.. Is th forc (magnitud) on th lctron lss than, qual to or gratr than th forc on th roton? 3. Is th acclration (magnitud) on th lctron lss than, qual to or gratr than th acclration of th roton? 4. Th tim intrval for th lctron to travl 5. mm is lss than, qual to or gratr than th tim intrval for th roton to travl 5. mm? 5. Is th work don on th lctron b th lctric forc lss than, qual to or gratr than th work don on th roton? 6. Is th kintic nrg of th lctron lss than, qual to or gratr than th kintic nrg of th roton? 7. Is th momntum of th lctron lss than, qual to or gratr than th momntum of th roton? 8. Has th otntial of th lctron dcrasd, sta th sam or incrasd? Has th otntial of th roton dcrasd, sta th sam or incrasd? 9. Is th otntial nrg of th lctron-fild lss than, qual to or gratr than th otntial nrg of th roton-fild? 11
12 Vrif ach of our rdictions b calculating ach quantit. Comar our rdictions with th numrical rsults and rsolv an discrancis. Solution Visualiz th motion of th two chargs: Th lctron will mov towards th ositiv lat as it is attractd to th ositiv lat and b rlld from th ngativ lat. Th oosit is tru for th roton, it will mov towards th ngativ lat. 1
13 Annotatd diagram including fram of rfrnc 1 Sinc th chargd articls start from rst, th will mov in a dirction aralll to th lctric fild lins. So, th distanc travlld corrsonds to th vrtical dislacmnts of th two chargs. Initial osition of lctron s mm s mm Final osition of lctron s mm s 5. mm 13
14 Initial osition of roton s mm s mm Final osition of roton s mm s 5. mm Th lctric forc on a chargd articl in an lctric fild is F q E Th magnituds of th chargs on th lctron and roton ar qual, thrfor th magnituds of th forcs on ach chargd articl ar qual. E E ˆj V 1 E V.m 5. 1 V.m 3 d 1 Elctric forc on lctron and roton FE q E Elctron N 8. 1 N F 8. 1 j N 15 ˆ Proton F 8. 1 j N 15 ˆ 3 Th acclration of a articl is givn b Nwton s Scond Law 1 a m F Th mass of th roton is much gratr than th mass of th lctron. So, th lctron s acclration is much gratr than th acclration of th roton. m / m /
15 Elctron Proton a a F ˆj m.s m 15 - F ˆj m.s m 1-4 Th acclration of th lctron is gratr than th acclration of th roton. So, th lctron will gain sd mor quickl and covr a givn distanc in a shortr tim intrval than th roton. Uniform acclration Elctron 1 s ut at v u at u s a t t m.s 5. 1 m 8.81 m.s? s v Proton s a s a t 9.41 m.s 6-1 u s a t t m.s 5. 1 m 4.81 m.s? s s a s v a t. 1 m.s
16 5 Work W F ds Wnt K Th sam work is don on th lctron and th roton b th lctric forc sinc th magnitud of th dislacmnts ar th sam. Elctron Proton W F s q E s J 17 W F s 4. 1 J 6 Th initial kintic nrgis of th lctron and roton ar both zro, hnc, W K K K mv Th final kintic nrgis will b th sam sinc th sam amount of work is don on th lctron and roton. Elctron 1 K W J K Proton 1 17 m 4. 1 J v altrnativ calculation K W J K 1 17 m v 4. 1 J altrnativ calculation 16
17 7 Momntum mv m K K m Elctron K K m m Proton m m m v kg.m.s 4-1 m v kg.m.s -1 dv 8 Potntial V E ˆj V E d d Th otntial is a linar function of distanc for a constant lctric fild. Th otntial diffrnc btwn th lats is 1 V, thrfor th otntial of th ositiv lat is +5 V and th otntial of th ngativ lat is -5 V. Th otntial at = 5. mm is V = 5 V and th otntial at = - 5. mm is V = -5 V. An lctron in an lctric fild movs from a rgion of lowr otntial to highr otntial. A roton movs from th highr otntial rgion to a lowr otntial rgion. 17
18 9 Potntial nrg U Th changs in otntial nrg of th lctron and th roton ar th qual sinc th work don on th chargs is th sam. U U W J Altrnativl, nrg must b consrvd. As th kintic nrg of th lctron or roton incrass thn th otntial nrg must dcras such that K U U K J W will now considr th [D] motion of a ositiv chargd articl in a uniform lctric fild. Th lctric fild is assumd to b uniform in th rgion btwn two oositl chargd aralll lats. Th otntial diffrnc btwn th two lats is V and th lat saration distanc is d. So, th uniform lctric fild btwn th lats is V E ˆj uniform lctric fild d 18
19 Fig. 3. Projctil motion if a chargd articl in a uniform lctric fild. Th quations dscribing th motion of a chargd articl in a uniform lctric fild ar idntical as thos dscribing th motion of an objct in th uniform gravitational fild nar th Earth s surfac as givn abov, ct th acclration of th chargd articl is ˆ q E ˆ q E a a i a j j a a m m 19
20 For th motion of th chargd articl w can also dfin th vr imortant trms otntial V and otntial diffrnc Th otntial and otntial diffrnc btwn two oints is rlatd to th diffrnc in otntial nrg of th lctric fild and charg. V. Elctric forc acting on charg FE q E ˆj Th onl forc acting on th articl is th lctric forc. So, th nt work don b th lctric forc is th work don b th lctric forc which rsults in th incras in kintic nrg whn th chargd articl is rojctd into th uniform lctric fild. Th lctric forc is a consrvativ forc, thrfor, th incras in kintic nrg rsults in a corrsonding dcras in th otntial sstm of th sstm (articl and charg).
21 Th initial osition of th chargd articls is takn as th Origin O(,) and is th rfrnc oint whr th otntial nrg and otntial ar both zro, U V. Elctric fild E E ˆj Elctric forc F q E ˆj q E E E Work don on articl W FE d s q E s Potntial nrg U W q E s U Potntial V E s q Kintic nrg K W mv mu q E s 1 1 K mv mv mu mu K mv mu mv mu
22 Prdict Elain Obsrv Eaml A articl of mass 1. kg and charg 1. mc is launchd with an initial vlocit of 1. m.s -1 at an angl of 75 o w.r.t. th horizontal in a uniform lctric fild V.m -1. Calculat th tim, vlocit, osition, kintic nrg, otntial nrg and total nrg of th articl at th Origin, whn it rachs its osition of maimum vrtical dislacmnt and whn it vrtical dislacmnt is again zro. Prdict th sha of th grahs for: Trajctor (s vs s ) Tim grahs a, a v, v s, s K, U, E Mark on th grahs, whn th articl rachs its maimum hight and rturns to its launch hight.
23 Solution q C m 1. kg u 1. m. s 75 m.s - 1 o E 1. 1 V.m a a qe m.s 1 m.s m Th initial vlocit and acclration valus ar idntical to thos givn in Eaml 1. Thrfor, th numrical valus and grahs ar idntical to th rsults rsntd in Eaml 1. In Eaml 1, th acclration of th articl is indndnt of th mass if ou launchd anothr articl with th sam initial conditions, but with diffrnt mass, thn all th grahs and numrical valus would b th sam. This is not tru for our articl launchd in a uniform lctric fild. Th acclration dos dndnt uon th articl s mass. 3
24 Anothr imortant diffrnc btwn th motion of articls in gravitational filds or lctric filds is: Th gravitational forc is alwas attractiv a articl undr th action of onl th gravitational forc will mov from a oint of high otntial nrg towards oints of lowr otntial nrg. Th lctric forc can b attractiv or rulsiv. A ositiv charg will mov from a highr otntial oint to a lowr otntial oint b th action of th lctric forc. Howvr, a ngativ charg will mov from a lowr otntial oint to a highr otntial oint b th action of th lctric forc. Considr th cas whr th vrtical acclration has a magnitud of 5. m.s - instad of 1 m.s -. This could b for th rojctil motion of an objct on a diffrnt lant or th motion of a chargd articl in an lctric fild. Using th rsults for Eaml 1, how do th trajctor and tim to rach maimum hight chang with th rducd vrtical acclration? 4
25 Comar th two sts of rsults for th vrtical acclration of 5. m.s - and 1. m.s -. 5
26 3 Eaml A bam of lctrons moving in th +X dirction with an initial vloict v ntrs a rgion of uniform lctric fild E dirctd in th +Y dirction gnratd b a air of oositl chargd aralll lats (aralll lat caacitor) which has a lngth C in th X dirction. Dtrmin th ath of th lctron whn it is moving through th uniform lctric fild. What is th vrtical distanc C th lctron will b dflctd whn assing through th uniform lctric fild. Dscib th ath of th lctron bam aftr laving th uniform lctric fild. Th lctron bam hits a scrn locatd at a distanc S from th nd of th aralll lat caacitor. What is th vrtical dflction S of th lctron bam from its original trajctor whn it striks th scrn? Solution Watch Vido (concntrat on th motion of chargd articls through a uniform lctric fild). Rviw motion with a uniform acclration Rviw [D] motion in a lan 6
27 Think about how to aroach th roblm b visulaizing th hsical situation. Draw an annoatd scintifc diagram. Stat th known and unknown hsical quantitis. Stat th quations that ou will nd. Stat th hsical rincils and concts ndd to solv th roblm. This is a long (difficult???) roblm, but if think about braking th roblm into four smallr roblms, thn ou will find that it is not such a difficult roblm. Th roblm can b solvd using th quations for uniform acclratd motion rsolvd into th X dirction and th Y dirction. v v a t s v t a t 1 v v a s uniform acclratd motion 7
28 X motion through caacitor (uniform lctric fild rgion) Initial conditions t v v a Transit tim s v t s t? t v C C Y motion through caacitor (uniform lctric fild rgion) Forc acting on lctron and its acclration F E m a E E a m 8
29 Initial conditions E t v a m Tim at which lctron lavs lctric fild rgion C t v Vrtical vlocit of lctron laving lctric fild v v at v E E C C m v mv Vrtical dislacmnt of lctron laving lctric fild s v t at 1 1 C C m v C E mv C E X motion fild fr rgion Initial conditions t v v a Tim for lctron to travl from caacitor to scrn s v t s t? t v S S 9
30 Y motion fild fr rgion Initial conditions E t v v C a mv Vrtical dislacmnt for osition of lctron striking scrn vt S S S C E E C v t mv C C S mv v E C E C S mv m v v E C C S mv Th vrtical dflction S of th lctron on th scrn is roortional to th lctric fild strngth btwn th lats. In cathod ra tubs usd in tlvision sts of th ast, th ath of th lctrons through th tub to th scrn could b controlld b changing th lctric fild btwn th lats of th caacitors. 3
31 VISUAL PHYSICS ONLINE htt:// If ou hav an fdback, commnts, suggstions or corrctions las mail: Ian Coor School of Phsics Univrsit of Sdn 31
7.4 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 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 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 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 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 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 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 informationJOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH 241) Final Review Fall 2016
JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw Fall 06 Th Final Rviw is a starting point as you study for th final am. You should also study your ams and homwork. All topics listd in th
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 informationPair (and Triplet) Production Effect:
Pair (and riplt Production Effct: In both Pair and riplt production, a positron (anti-lctron and an lctron (or ngatron ar producd spontanously as a photon intracts with a strong lctric fild from ithr a
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 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 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 informationUNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B. 3) Form the partial differential equation by eliminating the arbitrary functions
UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B 1) Form th artial diffrntial quation b liminating th arbitrar functions f and g in z f ( x ) g( x ) ) Form th artial diffrntial quation b liminating th arbitrar
More informationMassachusetts Institute of Technology Department of Mechanical Engineering
Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our
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 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 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 information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
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 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 informationContemporary, atomic, nuclear, and particle physics
Contmporary, atomic, nuclar, and particl physics 1 Blackbody radiation as a thrmal quilibrium condition (in vacuum this is th only hat loss) Exampl-1 black plan surfac at a constant high tmpratur T h is
More informationBETA DECAY VISUAL PHYSICS ONLINE
VISUAL PHYSICS ONLINE BETA DECAY Suppos now that a nuclus xists which has ithr too many or too fw nutrons rlativ to th numbr of protons prsnt for stability. Stability can b achivd by th convrsion insid
More informationSundials and Linear Algebra
Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.
More informationde/dx Effectively all charged particles except electrons
de/dx Lt s nxt turn our attntion to how chargd particls los nrgy in mattr To start with w ll considr only havy chargd particls lik muons, pions, protons, alphas, havy ions, Effctivly all chargd particls
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 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 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 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 informationorbiting electron turns out to be wrong even though it Unfortunately, the classical visualization of the
Lctur 22-1 Byond Bohr Modl Unfortunatly, th classical visualization of th orbiting lctron turns out to b wrong vn though it still givs us a simpl way to think of th atom. Quantum Mchanics is ndd to truly
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 informationDetermination of Vibrational and Electronic Parameters From an Electronic Spectrum of I 2 and a Birge-Sponer Plot
5 J. Phys. Chm G Dtrmination of Vibrational and Elctronic Paramtrs From an Elctronic Spctrum of I 2 and a Birg-Sponr Plot 1 15 2 25 3 35 4 45 Dpartmnt of Chmistry, Gustavus Adolphus Collg. 8 Wst Collg
More informationGradebook & Midterm & Office Hours
Your commnts So what do w do whn on of th r's is 0 in th quation GmM(1/r-1/r)? Do w nd to driv all of ths potntial nrgy formulas? I don't undrstand springs This was th first lctur I actually larnd somthing
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 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 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 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 informationDual Nature of Matter and Radiation
Higr Ordr Tinking Skill Qustions Dual Natur of Mattr and Radiation 1. Two bas on of rd ligt and otr of blu ligt of t sa intnsity ar incidnt on a tallic surfac to it otolctrons wic on of t two bas its lctrons
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 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 informationSPH4U Electric Charges and Electric Fields Mr. LoRusso
SPH4U lctric Chargs an lctric Fils Mr. LoRusso lctricity is th flow of lctric charg. Th Grks first obsrv lctrical forcs whn arly scintists rubb ambr with fur. Th notic thy coul attract small bits of straw
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
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 informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationREGISTER!!! The Farmer and the Seeds (a parable of scientific reasoning) Class Updates. The Farmer and the Seeds. The Farmer and the Seeds
How dos light intract with mattr? And what dos (this say about) mattr? REGISTER!!! If Schrödingr s Cat walks into a forst, and no on is around to obsrv it, is h rally in th forst? sourc unknown Phys 1010
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
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 informationClassical Magnetic Dipole
Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg
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 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 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 informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
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 informationAdditional Math (4047) Paper 2 (100 marks) y x. 2 d. d d
Aitional Math (07) Prpar b Mr Ang, Nov 07 Fin th valu of th constant k for which is a solution of th quation k. [7] Givn that, Givn that k, Thrfor, k Topic : Papr (00 marks) Tim : hours 0 mins Nam : Aitional
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 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 informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
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 informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
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 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 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 informationConstants and Conversions:
EXAM INFORMATION Radial Distribution Function: P 2 ( r) RDF( r) Br R( r ) 2, B is th normalization constant. Ordr of Orbital Enrgis: Homonuclar Diatomic Molculs * * * * g1s u1s g 2s u 2s u 2 p g 2 p g
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 informationAtomic energy levels. Announcements:
Atomic nrgy lvls Announcmnts: Exam solutions ar postd. Problm solving sssions ar M3-5 and Tusday 1-3 in G-140. Will nd arly and hand back your Midtrm Exam at nd of class. http://www.colorado.du/physics/phys2170/
More information1 Input-Output Stability
Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical
More informationRadiation Physics Laboratory - Complementary Exercise Set MeBiom 2016/2017
Th following qustions ar to b answrd individually. Usful information such as tabls with dtctor charactristics and graphs with th proprtis of matrials ar availabl in th cours wb sit: http://www.lip.pt/~patricia/fisicadaradiacao.
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 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 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 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 informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationMATHEMATICS (B) 2 log (D) ( 1) = where z =
MATHEMATICS SECTION- I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLY-ONE is corrct. Lt I d + +, J +
More informationME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
3.4 Forc Analysis of Linkas An undrstandin of forc analysis of linkas is rquird to: Dtrmin th raction forcs on pins, tc. as a consqunc of a spcifid motion (don t undrstimat th sinificanc of dynamic or
More informationPLASMA PHYSICS VIII. PROCESSING PLASMAS
PLASMA PHYSICS VIII. PROCESSING PLASMAS Introduction Plasmas ar usd to manufactur smiconductors, to modify th surfacs of matrials, to trat missions and wasts bfor thy ntr th nvironmnt, tc. Th plasma is
More informationTEMASEK JUNIOR COLLEGE, SINGAPORE. JC 2 Preliminary Examination 2017
TEMASEK JUNIOR COLLEGE, SINGAPORE JC Prliminary Eamination 7 MATHEMATICS 886/ Highr 9 August 7 Additional Matrials: Answr papr hours List of Formula (MF6) READ THESE INSTRUCTIONS FIRST Writ your Civics
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 informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
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 informationPrecise Masses of particles
/1/15 Physics 1 April 1, 15 Ovrviw of topic Th constitunts and structur of nucli Radioactivity Half-lif and Radioactiv dating Nuclar Binding Enrgy Nuclar Fission Nuclar Fusion Practical Applications of
More informationATMO 551a Homework 6 solutions Fall 08
. A rising air parcl in th cor of a thundrstorm achivs a vrtical vlocity of 8 m/s similar to th midtrm whn it rachs a nutral buoyancy altitud at approximatly 2 km and 2 mb. Assum th background atmosphr
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More information15. Stress-Strain behavior of soils
15. Strss-Strain bhavior of soils Sand bhavior Usually shard undr draind conditions (rlativly high prmability mans xcss por prssurs ar not gnratd). Paramtrs govrning sand bhaviour is: Rlativ dnsity Effctiv
More informationForces. Quantum ElectroDynamics. α = = We have now:
W hav now: Forcs Considrd th gnral proprtis of forcs mdiatd by xchang (Yukawa potntial); Examind consrvation laws which ar obyd by (som) forcs. W will nxt look at thr forcs in mor dtail: Elctromagntic
More informationElectromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Elctromagntic scattring Graduat Cours Elctrical Enginring (Communications) 1 st Smstr, 1388-1389 Sharif Univrsity of Tchnology Contnts of lctur 8 Contnts of lctur 8: Scattring from small dilctric objcts
More informationCollisions. In had on lastic collision of two bodis of qual ass ) Th fastr body spds up furthr and th slowr body slows down. ) Th fastr body slows down and th slowr body spds up. 3) Both of th abov. 4)
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationAnswer Homework 5 PHA5127 Fall 1999 Jeff Stark
Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw
More information5 Curl-free fields and electrostatic potential
5 Curl-fr filds and lctrstatic tntial Mathmaticall, w can gnrat a curl-fr vctr fild E(,, ) as E = ( V, V, V ), b taking th gradint f an scalar functin V (r) =V (,, ). Th gradint f V (,, ) is dfind t b
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 informationu x A j Stress in the Ocean
Strss in t Ocan T tratmnt of strss and strain in fluids is comlicatd and somwat bond t sco of tis class. Tos rall intrstd sould look into tis rtr in Batclor Introduction to luid Dnamics givn as a rfrnc
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationUnit 7 Charge-to-mass ratio of the electron
Unit 7 Charg-to-ass ratio of th lctron Kywords: J. J. Thoson, Lorntz Forc, Magntic Filds Objctiv: Obsrv th rsults of lctron ba influncd by th agntic fild and calculat th charg-to-ass ratio of th lctron.
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 informationSchrodinger Equation in 3-d
Schrodingr Equation in 3-d ψ( xyz,, ) ψ( xyz,, ) ψ( xyz,, ) + + + Vxyz (,, ) ψ( xyz,, ) = Eψ( xyz,, ) m x y z p p p x y + + z m m m + V = E p m + V = E E + k V = E Infinit Wll in 3-d V = x > L, y > L,
More informationANSWERS C C =
107 CHAPTER E1 ANSWERS 1.2 No. of xcss lctrons -0.1 10-6 C -1.6 10-19 C 6.2 1011 6.2 10 11 lctrons hav bn transfrrd from your hair to th comb. 1.3 Elctrical: 1, 2, 3, 4, 6, 8, 12, 13 Non-lctrical: 5, 11
More informationIntro to Nuclear and Particle Physics (5110)
Intro to Nuclar and Particl Physics (5110) March 09, 009 Frmi s Thory of Bta Dcay (continud) Parity Violation, Nutrino Mass 3/9/009 1 Final Stat Phas Spac (Rviw) Th Final Stat lctron and nutrino wav functions
More information(A) (C) relation for the Legendre polynomial is α given by Pm. (A) σ = m. (B) σ 2 = m (C) σ + m = 0 (D) σ = m
. h atrix i Only Hritian i is Only Unitary Hritian and Unitary Nithr Hritian nor Unitary. What is th product of ign valus of 6. h first proprty of th orthogonality rlation for th Lgndr polynoial is α 0
More informationAP Calculus BC AP Exam Problems Chapters 1 3
AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum
More information