11 Oct 18 Millika.1 MILLIKAN OIL DROP EXPERIMENT This xprimt is dsigd to show th quatizatio of lctric charg ad allow dtrmiatio of th lmtary charg,. As i Millika s origial xprimt, oil drops ar sprayd ito a rgio whr a uiform lctric fild ca b stablishd, ad th motios of drops ar studid udr th actio of th lctric fild big turd o ad off. Although this xprimt will allow o to masur th total charg o a drop, it is oly through a aalysis of th data obtaid, ad a crtai dgr of xprimtal skill, that th lmtary charg ca b dtrmid. By slctig drops which ris ad fall slowly, o ca b crtai that th drop has a small umbr of xcss lctros. A umbr of such drops should b obsrvd ad thir rspctiv chargs calculatd. If th chargs o ths drops ar itgral multipls of a crtai smallst charg, th this is a good idicatio of th quatum atur of lctricity. Thory: A aalysis of th forcs actig o a chargd droplt will allow dtrmiatio of its charg. Figur 1 shows th forcs actig o th drop wh it is fallig i air ad has rachd its trmial vlocity (trmial vlocity is rachd i a fw milliscods for th droplts usd i this xprimt). I Figur 1 υf is th vlocity of fall, k is th cofficit of frictio btw th air ad th drop, m is th mass of th drop, ad g is th acclratio du to gravity. Sic th t forc o th drop is zro (costat vlocity motio), mg = kυf (1) + + + + + + + + + + kυf E mg kυr mg Figur 1 Figur 2
11 Oct 18 Millika.2 Figur 2 shows th forcs actig o th drop wh it is risig udr th ifluc of a lctric fild. E is th lctric fild, is th charg carrid by th drop, ad υr is th vlocity of ris. Addig th forcs vctorially (ad assumig that trmial vlocity has b rachd) yilds: E = mg + kυr (2) I both cass thr is also a small buoyat forc xrtd by th air o th droplt. Th buoyat forc is tak ito accout by substitutig ρ = ρoil ρair for ρoil whrvr th dsity of th oil occurs i subsqut quatios. Elimiatig k from quatios (1) ad (2) ad solvig for yilds: mg( υ +υ ) Ev f r = () f To limiat m from quatio (), o uss th xprssio for th volum of a sphr: m = 4 πa ρ (4) whr a is th radius of th droplt ad ρ is th oil dsity (corrctd for buoyacy). To calculat a o mploys Stok s Law, rlatig th radius of a sphrical body to its vlocity of fall i a viscous mdium, amly 9ηυf a = (5) 2gρ whr η is th cofficit of viscosity of air. Substitutig quatio (5) ito quatio (4), ad th substitutig for m i quatio () yilds 4 1 9η ( υf + υr ) υf = π (6) gρ 2 E Howvr, for vry small drops, th drop radii ar o th ordr of th itr-molcular spacig of th air i which th drops ar movig. Thus th assumptio, implicit i Stok s Law, that th air ca b tratd as a cotiuous mdium, is o logr valid. As dtrmid by Millika, a corrctio factor of 1+ b pa / 2 (7) must b icludd i th xprssio for to accout for th ihomogity of th air. I th corrctio factor, p is th atmosphric i cm of Hg, a is th radius of th drop (i m) as calculatd by th ucorrctd form of Stok s Law (quatio (5)), ad b = 6.17 10 6 m cm Hg, a costat.
11 Oct 18 Millika. Th lctric fild is giv by E = V/d (quatio (8)) whr V is th pottial diffrc across th paralll plats sparatd by a distac d. Substitutig for th lctric fild ad th corrctio factor i quatio (6) yilds: / 2 4 1 9η b ( υf + υr ) υf = πd 1 + ρg 2 pa V (9) Aothr mthod for dtrmiig th charg o th oil drop is to adjust th voltag (ad hc th lctric fild) util th oil drop floats. Th charg o th oil drop is th giv by whr U = float voltag. Apparatus: 1 / 2 / 2 4 1 9η b υf = πd + (10) ρg 2 pa U Th quipmt supplid, as show i Figur, cosists of a stad supportig th oil drop chambr, th light, ad th masurig microscop; ad a powr supply for th light ad th voltag to produc th lctric fild i th chambr. I additio, a lctroic digital stopwatch is providd for masurig th ris ad fall tims of th oil drops. A atomizr is usd to produc th oil drops. Th ozzl of th atomizr is placd agaist th two borhols of th chambr. A quick squirt will fill th chambr with drops which bcom visibl i th viwig ara. Wh dtrmiig th oil drop spds, th actual distac, s, travlld by a drop is giv by y s = 2 017 10 4. m (11) whr y is th umbr of microscop scal divisios through which th drop movs. (Ask your lab istructor to cofirm that 2.017 is th corrct umbr to us i this quatio.) NOTE: Th optics of th microscop caus imag ivrsio. Drops appar to ris udr gravity ad fall wh th lctric fild is o. For th apparatus providd: η = 1.8 10 5 N s/m 2 d = 6.00 10 m ρ = 874 kg/m
11 Oct 18 Millika.4 Figur Substitutig ths valus ito quatios (5), (9), ad (10) yilds a = 9.80 10 5 υf m (12) = 2.0 10 10 6 6.17 10 m cmhg 1+ pa / 2 ( υ +υ ) f V r υ f (1)
11 Oct 18 Millika.5 6 /2 10 6.17 10 m cmhg υf = 2.0 10 1+ pa (14) U Procdur ad Exprimt: Rcord th atmosphric prssur i cm Hg. All quatitis othr tha th atmosphric prssur ar to b xprssd i SI uits. For ach of 10 oil drops, masur 4 sts of FIELD ON (500 V) ad FIELD OFF tims ad distacs. (For FIELD ON th drops ar actually risig but appar to fall, for FIELD OFF th drops ar actually fallig but appar to ris.) Also, for ach drop, masur th float pottial. (Th float pottial is th applid voltag for which th drop has gligibl vrtical motio.) To sur that th drops ar small ad hav a small umbr of xcss chargs, choos drops which mov slowly both wh th fild is o ad wh it is off. For bst rsults, sur that som of th masurd drops ar movig at ris ad fall spds such that it taks about 15 scods to mov 10 microscop divisios with th fild o or off. It is also importat that som of th drops mov fastr tha this with th fild o or off, so that you obtai som drops with mor tha o xcss charg. Data for ach drop should b rcordd as follows: Drop Trial Tim (± s) 1 1 2 4 2 1 tc. FIELD ON Divisios (± ) / 2 FIELD OFF Tim (± s) Divisios (± ) Float Pottial (± V) Usig th avrag ris ad fall spds calculat th charg o ach drop. Usig th avrag fall spd ad th float pottial calculat th charg o ach drop. (Not that th subscripts i th quatios rfr to actual (rathr tha appart) drop motio.) A Excl spradsht is availabl to aid with th prstatio ad aalysis of th data. Not that you must do a complt st of sampl calculatios for o of th oil drops by had. Th spradsht is obtaid from th lab maual wb pag: http://physics.usask.ca/~bzulkosk/lab_mauals/ : 1. Dowload ad op th spradsht millika.xls. 2. Etr th tim rror i cll D4, th divisio rror i cll D5, ad th atmosphric prssur ad its rror i clls J4 ad L4.. Etr th oil drop data i th appropriat clls of colums C, D, H, I, M, ad O. 4. Rmmbr to priodically sav your work. (Slct Sav from th Fil mu.)
11 Oct 18 Millika.6 5. Bfor lavig th lab, hav your spradsht data ad rsults vrifid by o of th istructors. By carful aalysis of th rsults for ach mthod (ris/fall mthod ad float mthod), vrify th quatum atur of lctric charg (i.. th xistc of a lmtary charg). Calculat th valu of th lmtary charg ad compar to th accptd valu of 1.602 10 19 C. Rfrcs: R.A. Millika, Th Elctro, Chicago, Uivrsity of Chicago Prss L. Pag, Itroductio to Thortical Physics, Nw York, Va Nostrad, Chpt. 6