Coulomb Balance. Important Information. 1 Introduction

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1 Coulomb Balance Equipmen: Coulomb balance apparaus, 5 5 inch plasic plae, vernier calipers, power supply, se of fracional weighs (less han 500 mg) wih weezers, resisance box se a 1 MΩ, 5 leads, index cards, analog volmeer, laser on ripod, small conainer wih rods and humb screws (used o moun he lower capacior plae), and a 6in ruler Reading: Coulomb s Law in your ex book Imporan Informaion 1. HIGH VOLTAGE FROM A POWER SUPPLY IS APPLIED TO TWO CAPACITOR PLATES. DO NOT TOUCH THE WIRES PROVIDING THIS VOLTAGE. BE SURE THAT A 1 MΩ RESISTOR IS CONNECTED IN SERIES WITH THE CAPACITOR. 2. This experimen uses a laser beam as an opical lever arm. DO NOT LET THE LASER BEAM OR ITS REFLECTION ENTER YOUR EYE. SERIOUS DAMAGE TO YOUR EYE MAY RESULT. 1 Inroducion In SI unis he magniude of he force F beween wo charges q 1 and q 2 in vacuum is given by F = q 1q 2 4πɛ 0 R 2, (1) where R is he disance beween he wo charges and ɛ 0 is a consan called he vacuum of permiiviy. The uni of charge in SI unis is a coulomb. This equaion shows ha 1/ɛ 0 plays he role of a proporionaliy consan, which for any values of he charges gives he correc magniude and dimensions for he expression so ha he quaniy on he righ properly represens a force. The srengh of he elecrosaic ineracion beween charges. is relaed o he inverse of permiiviy. In his sense he permiiviy plays a role for elecrical forces which is analogous o ha played by he graviaional consan in Newon s expression for he graviaional force beween wo masses. The experimen o be performed wih a Coulomb Balance is o measure he force beween wo separaed charged objecs having a known volage beween hem and having a known configuraion. From he daa, he value of ɛ 0 can be deduced. The experimen is he elecrical analog of he Cavendish experimen for graviaional forces, which measures he graviaional force beween wo known masses. 1

2 Maxwell showed in he mid-19h cenury ha by combining his equaions conaining he elecric and magneic fields he could predic he exisence of a propagaing wave conaining boh ypes of fields. He prediced ha he speed c of he wave in vacuum is given by c = 1 ɛ0 µ 0. (2) The new consan µ 0 is called he vacuum permeabiliy. I appears in he expression for he force beween wo elecrical currens separaed by vacuum and is a magneic effec. The prediced value for c, approximaely meers per second, has proved o be idenical o he value obained by Michelson for he speed of ligh, hereby supporing he heory ha ligh is an example of elecromagneic radiaion. Consequenly, by measuring ɛ 0 and µ 0 we have an indirec way of deermining c, he speed of ligh. This experimen is concerned wih measuring he permiiviy ɛ 0. In a laer experimen he permeabiliy µ 0 will be examined. 2 Descripion of he Experimen and Commens This is a difficul experimen (no everyhing in life is easy), and you should no be unduly upse if your resuls differ subsanially from he acceped value! See Fig. 1 and Fig. 2 on how he apparaus is assembled. In he apparaus, a parallel plae capacior has one fixed horizonal plae. The oher plae pivos above he fixed horizonal plae so ha he separaion d beween he plaes is variable. The posiion of he movable plae can be accuraely moniored by a laser beam refleced off a mirror aached o he movable plae assembly. A small weigh is placed in he cener of he movable plae and he posiion of he op plae is hen deermined. The weigh is removed which causes he plaes o separae a bi. A volage is applied beween he plaes so ha he plae separaion is he same as when he weigh was in place. The applied volage is now producing a force of aracion beween he plaes ha is equal o he weigh. A simple equaion relaes he force, volage, plae area, and plae separaion o he permiiviy of space. 3 Theory Consider a parallel plae capacior whose dielecric is vacuum or air, which we consider o be equivalen for he purposes of his experimen. If a volage V is mainained beween he plaes by a power supply he plaes will have charges ha are equal in magniude and opposie in sign. The charges of opposie polariies on he plaes causes an aracive force beween he plaes. This force, for plaes of equal area, is given by where F = ɛ 0AV 2 2d 2, (3) F is he force of aracion in newons, A is he area of each plae in square meers, 2

3 V is he poenial difference beween he plaes in vols, and d is he disance separaing he plaes in meers. In his experimen everyhing in Eq. (3) is measured or deermined excep ɛ 0. The analysis of he resuls will be based on Eq. (3) wrien in he form 4 The Apparaus ɛ 0 = 2F d2 AV 2. (4) Fig. 1 is a phoograph of he Coulomb Balance and Fig. 2 is a side skech of he enire se-up in cross-secion. During he experimen, he boom capacior plae is normally fixed in posiion. This plae is held by wo rods ha are secured by four humbscrews. This allows he boom plae o be roaed and iled if necessary. The op capacior plae is held by wo rods ha are aached o a pivo arm ha pivos on wo knife edges. This allows he op plae o freely roae oward and away from he boom plae. This op plae assembly has a couner weigh or posiion adjusmen weigh ha allows he verical posiion of he op capacior plae o be adjused. There is a mirror aached o he pivo arm. A laser beam is direced ono he mirror and hen refleced back o a piece of paper aped o he fron of he laser. This allows you o reurn he op plae o a given posiion wih grea accuracy. The period adjusmen weigh changes he oscillaion period of he op plae. Moving his weigh down makes he period of oscillaion longer, and lenghens he imes necessary for he op plae o sop oscillaing. This weigh, being below he pivo poins, also serves o make he posiion of he op plae sable. Sabiliy of he op plae increase as you he lower his weigh. There is a cenering rod (no shown) ha is a rod wih wo knobs a he ends and wo off-se poins. When he cenering rod is roaed, he wo poins fi ino wo apered holes in he boom of he pivo arm, and one can lif he pivo arm a bi. When he pivo arm is lowered back down, he knife edges should be cenered on heir suppors. This should be done genly so as no o damage he knife edges. There is a plane plasic plae he size of he capacior plaes. This is emporarily insered beween he capacior plaes so ha he wo plaes can be se parallel o one anoher a a fixed separaion d, where d is he hickness of he plasic plae. Wih he plasic plae insered and he plaes pressed ogeher, he laser beam is used mark he posiion of he upper plae so ha his posiion can be recovered afer he plasic plae has been removed. A meal plae aached o he pivo arm is posiioned in he gap of a small permanen magne. As he pivo arm swings back and forh, his arrangemen provides eddy curren damping. 5 Apparaus Se-Up This apparaus is sensiive! Once you sar adjusing i and aking daa, don lean on he bench or breah in he direcion of he plaes, and be careful no 3

4 PHYS-UA 72 Inro o Exp Physics II Coulomb Balance o bump he able. Wach for air currens, and since he power supply has a fan, don pu i near he apparaus. I is all oo easy o sar he op plae oscillaing, and as you will find ou, seing he op plae ino oscillaion is a bad idea. 1. Check ha he board on which he apparaus is mouned does no wobble. If i does, adjus one of he wo humbscrews on he fron of he board. 2. Measure and record he lengh and widh of he wo capacior plaes. I is no necessary o measure he hickness of he capacior plaes. 3. Wire up he apparaus, power supply, and volmeer as shown in Fig. 2. Leave he power supply off and he volage conrol knob fully CCW. 4. Moun he lower and upper capacior plaes as shown in Fig Try seing he period adjusmen weigh oward he lower end of is range. If you decide o move i, you will have o realign he apparaus as described below. 6. A capacior plae pivos on wo knife edges, which res on fla surfaces. Boh he knife edges and fla surfaces are easily damaged. Please handle hem wih care, using he cenering rod, described below, o cener he knife edges and o genly lower he knife edges ono he fla surfaces 7. Cener he pivo arm by genly raising i and lowering i wih he cenering rod. 8. Check ha he op capacior plae oscillaes freely. The damping plae should no ouch he magne, and he poins on he cenering rod should no ouch he pivo arm. A his poin look above he capacior plaes and check ha he plaes are perfecly aligned o one anoher. If no go back adjus he plaes. 4

5 9. Measure and record he hickness of he plasic plae. 10. Pu he plasic plae beween he capacior plaes, pu a penny in he cener of he op plae o force i down ono he plasic plae, and check ha he op plae lies fla on he plasic. If i does no, loosen he four screws holding he boom plae and make he wo plaes as parallel as possible. A his poin you will probably observe ha he capacior plaes are no perfecly fla. Make an esimae as o how much you hink he deviaions from flaness are. The hickness of he plasic will be he minimum disance beween he plaes. The average disance beween he plaes will be more han his. Leave he plasic plae in place and he penny on he op plae. 11. WITH THE LASER OFF, familiarize yourself wih he knobs on he ripod moun so ha you can raise, lower, and roae he laser wih ease. In adjusing he laser beam, you may also have o move he ripod sideways. 12. Tape a piece of paper o he fron of he laser and o one side of he laser beam opening. REMINDER: DO NOT LET THE LASER BEAM OR ITS REFLECTION ENTER YOUR EYE. Turn on he laser, open he shuer, and adjus he posiion and orienaion of he laser beam so ha i reflecs off he mirror and back ono he paper aped o he laser. Make a horizonal mark where he laser beam srikes he paper. You migh consider making he mark no a he cener of he laser beam, bu a he op or boom of he laser beam. 13. Remove he penny and he plasic plae, aking care no o disurb he posiion of he op plae. Place a 50 mg mass in he cener of he op plae. Roae he posiion adjusmen weigh unil he laser beam his he mark you have already drawn on he paper aached o he laser. The op plae should now be in he same posiion i was in wih he plasic beween he plaes. 14. Remove he 50 mg mass, aking care no o disurb he apparaus. The capacior plaes will swing apar o a new equilibrium posiion. 15. Check your wiring and ha he resisance box is se for 1 MΩ. Leave i a ha value hroughou he experimen. I is more han likely ha he capacior plaes will shor (ouch) several imes during he experimen. The 1 MΩ resisor will limi he curren o a safe value. Noe. If you are using an exernal volmeer ha has a sensiiviy swich, you should sar wih he swich in he leas sensiive posiion so as o proec he meer. Then choose he mos sensiive scale ha keeps he needle of he meer on scale. If your volmeer has more han one se of numbers on is scale, choose he numbers ha correspond o he full scale value as given by he sensiiviy swich, adjusing he decimal poin as necessary. 5

6 6 Procedures Se your volmeer on he leas sensiive scale. Turn on he power supply. Saring from zero, increase he volage beween he capacior plaes. This will bring he plaes oward one anoher. Your goal is o swing he op plae so ha he refleced laser beam his he mark on he paper aached o he laser, and record he volage V necessary o do his. The ask is very difficul because as he plaes ge closer ogeher, he siuaion ges unsable. This is due o he fac ha even if he volage is held consan, he force varies as 1/d 2. If he op plae, due o some perurbaion, moves oward he lower plae, he Coulomb aracive force beween he plae increases, and may be larger han he resoring force supplied by he period adjusmen weigh. The op plae will accelerae ino he boom plae. Hence he need for he 1 MΩ resisor. As he plaes approach he desired separaion, increase he volage very slowly so as no o give he op plae oo much momenum oward he lower plae. If you have rouble, ry moving he period adjusmen weigh o a new posiion. If you do his, you will have o realign he apparaus. Make a number of runs, obaining muliple values of he necessary volage V ha will resul in he plaes being a he same posiion as wih he 50 mg on he op plae. A ha volage, he Coulomb force beween he plaes will be he weigh of he 50 mg mass. Use Eq. (4) o obain muliple values of ɛ 0. Compare your values o he acceped value. 7 Addiional Quesions 1. In placing he 50 mg mass on he op plae, you are advised o place his mass in he cener of he op plae. Would you ge he same resul if he mass were no placed in he cener, bu perhaps closer or furher away from he balance beam? Discuss. 2. Idenify and discuss wha you consider o be he main sources of error. Could you sugges how some of hese errors migh be reduced? 3. There is an insabiliy in he experimen in ha he force beween he plaes increases slighly wih decreasing disance beween he plaes (why do you hink his is? 8 Commen The Coulomb and Graviaional forces are similar in ha heir srengh varies inversely wih he square of he disance. They differ in ha he Coulomb force can be repulsive as well as aracive, and in ha he Coulomb force beween elemenary paricles is orders of magniude sronger han he graviaional force. In analyzing he hydrogen aom and oher aoms, he graviaional force is oally ignored. 9 Finishing Up Please leave he bench as you found i. Thank you. 6

a es kni f e edge per i od adj us men wei gh magne i c dampi ng devi ce Res

7 PHYS-UA 72 Inro o Exp Physics II Coulomb Balance Figure 1: Coulomb Balance mi r r or l as erbeam pi voar m l as er d = gap s paci ng paper capaci or pl a es kni f e edge per i od adj us men wei gh magne i c dampi ng devi ce Res i s ance Box( 1MΩ) Power Suppl y + com Figure 2: Seup Cross Secion 7 Vol me er +

Constant Acceleration

Constant Acceleration Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion