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
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
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
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
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 1 1 1 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
W K mv mu mv mu U W m g s 1 1 1 1 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
Solution Us th grahs shown blow to chck our calculations and rdictd grahs. 7
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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
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
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
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
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 -1 4-1 q E Elctron 19 4 15 1.6 1 5. 1 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 7 31 3 / m 1.67 1 / 9.111 1.833 1 14
Elctron Proton a a F 8.8 1 ˆj m.s m 15 - F 4.8 1 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 -1 3 15 - m.s 5. 1 m 8.81 m.s? s v Proton s a 9 1.1 1 s a t 9.41 m.s 6-1 u s a t t -1 3 1 - m.s 5. 1 m 4.81 m.s? s s a 8 4.6 1 s v a t. 1 m.s 5-1 15
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 17 4. 1 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 17 4. 1 J K Proton 1 17 m 4. 1 J v altrnativ calculation K W 17 4. 1 J K 1 17 m v 4. 1 J altrnativ calculation 16
7 Momntum mv m K K m Elctron K K m m Proton m m m v 8.5 1 kg.m.s 4-1 m v 3.7 1 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
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 17 4. 1 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 17 4. 1 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
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
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).
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 1 1 1 1 K mv mu mv mu 1 1 1 1 1
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 1.1 5 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.
Solution q 3 1. 1 C m 1. kg u 1. m. s 75 m.s - 1 o E 1. 1 V.m a a 5-1 3 5 1. 1 1. 1 qe m.s 1 m.s m 1 - - 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
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
Comar th two sts of rsults for th vrtical acclration of 5. m.s - and 1. m.s -. 5
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
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
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
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
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
VISUAL PHYSICS ONLINE htt://www.hsics.usd.du.au/tach_rs/hs/s/shom.htm If ou hav an fdback, commnts, suggstions or corrctions las mail: Ian Coor School of Phsics Univrsit of Sdn ian.coor@sdn.du.au 31