Revision Guide for Chapter 16

Transcription

1 Revision Guide fo Chapte 16 Contents Revision Checklist Revision Notes lectic field... 5 Invese squae laws... 7 lectic potential... 9 lecton... 9 Foce on a moving chage Mass and enegy... 1 Relativistic calculations of enegy and speed lecton volt Summay Diagams The electic field between paallel plates Two ways of descibing electic foces Field stength and potential gadient Field lines and equipotential sufaces... 1 Invese squae law and flux... Radial fields in gavity and electicity... 3 How an electic field deflects an electon beam... 5 Foce, field, enegy and potential... 6 Millikan s expeiment... 7 How a magnetic field deflects an electon beam... 8 Foce on a cuent: foce on a moving chage... 9 Measuing the momentum of moving chaged paticles The ultimate speed Betozzi s demonstation Relativistic momentum p = γmv... 3 Relativistic enegy total = γmc negy, momentum and mass Advancing Physics A 1

2 Revision Checklist Back to list of Contents I can show my undestanding of effects, ideas and elationships by descibing and explaining cases involving: a unifom electic field = V /d (measued in volts pe mete) Revision Notes: electic field Summay Diagams: The electic field between paallel plates, Two ways of descibing electical foces, Field stength and potential gadient, Field lines and equipotential sufaces the electic field of a chaged body; the foce on a small chaged body in an electic field; the invese squae law fo the field due to a small (point o spheical) chaged object Revision Notes: electic field, invese squae laws Summay Diagams: Invese squae law and flux, Radial fields in gavity and electicity, How an electic field deflects an electon beam electical potential enegy and electic potential due to a point chage and the 1 / elationship fo electic potential due to a point chage Revision Notes: electic potential, invese squae laws Summay Diagams: Foce, field, enegy and potential, Radial fields in gavity and electicity evidence fo the disceteness of the chage on an electon Revision Notes: electon Summay Diagams: Millikan's expeiment the foce qvb on a moving chaged paticle due to a magnetic field Revision Notes: foce on a moving chage Summay Diagams: How a magnetic field deflects an electon beam, Foce on cuent: foce on moving chage, Measuing the momentum of moving chaged paticles elativistic elationships between mass and enegy Revision Notes: mass and enegy, elativistic calculations of enegy and speed Summay Diagams: The ultimate speed Betozzi's demonstation, Relativistic momentum, Relativistic enegy, negy, momentum and mass I can use the following wods and phases accuately when descibing effects and obsevations: electic chage, electic field; electic potential (J C 1 ) and electical potential enegy (J); equipotential suface Revision Notes: electic field, invese squae laws, electic potential Summay Diagams: Foce, field, enegy and potential the electon volt used as a unit of enegy Revision Notes: electon volt Advancing Physics A

3 I can sketch and intepet: gaphs of electic foce vesus distance, knowing that the aea unde the cuve between two points gives the electic potential diffeence between the points gaphs of electic potential and electical potential enegy vesus distance, knowing that the tangent to the potential vs distance gaph at a point gives the value of the electic field at that point Summay Diagams: Foce, field, enegy and potential, Radial fields in gavity and electicity diagams illustating electic fields (e.g. unifom and adial ) and the coesponding equipotential sufaces Revision Notes: electic field, invese squae laws Summay Diagams: Field stength and potential gadient, Field lines and equipotential sufaces I can make calculations and estimates making use of: the foce q on a moving chaged paticle in a unifom electic field Revision Notes: electic field Summay Diagams: How an electic field deflects an electon beam adial component of electic foce due to a point chage kqq F electic = adial component of electic field due to a point chage Felectic kq electic = = q Revision Notes: electic field, invese squae laws Summay Diagams: Foce, field, enegy and potential electic field elated to electic potential diffeence electic dv = dx and = V electic d fo a unifom field Revision Notes: electic field Summay Diagams: Foce, field, enegy and potential, Field stength and potential gadient electic potential at a point distance fom a point chage: V = kq Revision Notes: electic potential Summay Diagams: Foce, field, enegy and potential Advancing Physics A 3

4 the foce F on a chage q moving at a velocity v pependicula to a magnetic field B: F = q v B Revision Notes: foce on a moving chage Summay Diagams: How a magnetic field deflects an electon beam, Foce on cuent: foce on moving chage, Measuing the momentum of moving chaged paticles Advancing Physics A 4

5 Revision Notes Back to list of Contents lectic field lectic fields have to do with the foces electic chages exet on one anothe. lectic fields ae impotant in a wide ange of devices anging fom electonic components such as diodes and tansistos to paticle acceleatos. An electic field occupies the space ound a chaged object, such that a foce acts on any othe chaged object in that space. The lines of foce of an electic field tace the diection of the foce on a positive point chage. lectic field lines Nea a positive point chage lectic field lines Nea a negative point chage Advancing Physics A 5

6 lectic field lines Nea opposite point chages The electic field stength at a point in an electic field is the foce pe unit chage acting on a small positive chage at that point. lectic field stength is a vecto quantity in the diection of the foce on a positive chage. The SI unit of electic field stength is the newton pe coulomb (N C 1 ) o equivalently the volt pe mete (V m 1 ). The foce F on a point chage q at a point in an electic field is given by F = q, whee is the electic field stength at that point. If a point chage q is moved a small distance δ x along a line of foce in the diection of the line, the field acts on the chage with a foce q and theefoe does wok δ W on the chage equal to the foce multiplied by the displacement. Hence δ W = qδ x so the potential enegy P of the chage in the field is changed by an amount δ P = qδ x, whee the minus sign signifies a eduction. Since the change of potential is given by δ V = δ q P then δ V = δ x, so that δv =. δx The electic field is thus the negative gadient of the electic potential. In the limit δ x 0 dv =. dx Thus the lage the magnitude of the potential gadient, the stonge is the electic field stength. The diection of the electic field is down the potential gadient. A stong field is indicated by a concentation of lines of foce o by equipotential sufaces close togethe. Advancing Physics A 6

7 Foce on a point chage in a unifom field q F A unifom electic field exists between two oppositely chaged paallel conducting plates at fixed sepaation. The lines of foce ae paallel to each othe and at ight angles to the plates. Because the field is unifom, its stength is the same in magnitude and diection eveywhee. The potential inceases unifomly fom the negative to the positive plate along a line of foce. Fo pependicula distance d between the plates, the potential gadient is constant and equal to V / d, whee V is the potential diffeence between the plates. The electic field stength theefoe has magnitude = V / d. A point chage q at any point in the field expeiences a foce q V / d at any position between the plates. Relationships F = q gives the foce on a test chage q in an electic field of stength. dv =. dx = V / d fo the electic field stength between two paallel plates. Invese squae laws Radiation fom a point souce, and the electic and gavitational fields of point chages and masses espectively all obey invese squae laws of intensity with distance. An invese squae law is a law in which a physical quantity such as adiation intensity o field stength at a cetain position is popotional to the invese of the squae of the distance fom a fixed point. Advancing Physics A 7

8 The invese squae law souce sphee of adius The following quantities obey an invese squae law: The intensity of adiation I fom a point souce (povided the adiation is not absobed by mateial suounding the souce) is given by W I = 4π whee is the distance fom the souce and W is the ate of emission of enegy by the souce. The facto 4 π aises because all the adiation enegy emitted pe second passes though a sphee of suface aea 4 π at distance. The adial component of electic field stength at distance fom a point chage Q in a vacuum Q =. 4πε 0 The lines of foce ae adial, speading out fom Q. The invese squae law fo the intensity of the field shows that the lines of foce may usefully be egaded as continuous, with thei numbe pe unit aea epesenting the field intensity, since in this case the lines will cove the aea 4 π of a sphee suounding the chage. The adial component of gavitational field stength, g, at distance fom the cente of a sphee of mass M, GM g =. The lines of foce ae adial. As with the electic field of a point chage, the invese squae law means that the lines of foce may be thought of as continuous, epesenting the field intensity by thei numbe pe unit aea. Then the facto may be thought of as elated to the suface aea of a sphee of adius which the field has to cove. Relationships Radiation intensity I = W 4π Advancing Physics A 8

9 at distance fom a point souce of powe W. lectic field Q = 4πε 0 Gavitational field GM g =.. lectic potential The electic potential at a point is the potential enegy pe unit chage of a small positive test chage placed at that point. This is the same as the wok done pe unit positive chage to move a small positive chage fom infinity to that point. The potential enegy of a point chage q is p = q V, whee V is the potential at that point. The unit of electic potential is the volt (V), equal to 1 joule pe coulomb. lectic potential is a scala quantity. Potential gadient δx q F The potential gadient, dv / dx, at a point in an electic field is the ate of change of potential with distance in a cetain diection. The electic field stength at a point in an electic field is the negative of the potential gadient at that point: dv =. dx In the adial field at distance fom a point chage Q the potential V is: Q V =. 4πε 0 The coesponding electic field stength is: dv = d d Q = d 4πε Q = 4πε 0 0. lecton The electon is a fundamental paticle and a constituent of evey atom. The electon caies a fixed negative chage. It is one of six fundamental paticles known as leptons. Advancing Physics A 9

10 The chage of the electon, e, is C. The specific chage of the electon, e / m, is its chage divided by its mass. The value of e / m is C kg 1. The enegy gained by an electon acceleated though a potential diffeence V is ev. If its speed v is much less than the speed of light, then ev = (1/) mv. lectons show quantum behaviou. They have an associated de Boglie wavelength λ given by λ = h/p, whee h is the Planck constant and p the momentum. At speeds much less than the speed of light, p = mv. The highe the momentum of the electons in a beam, the shote the associated de Boglie wavelength. Relationships The electon gun equation (1 / ) m v = e V (fo speed v much less than the speed of light). Foce on a moving chage The foce F on a chaged paticle moving at speed v in a unifom magnetic field is F = q v B sinθ whee q is the chage of the paticle and θ is the angle between its diection of motion and the lines of foce of the magnetic field. The diection of the foce is pependicula to both the diection of motion of the chaged paticle and the diection of the field. The foces on positively and negatively chaged paticles ae opposite in diection. A beam of chaged paticles in a vacuum moving at speed v in a diection pependicula to the lines of a unifom magnetic field is foced along a cicula path because the magnetic foce q v B on each paticle is always pependicula to the diection of motion of the paticle. Foce on a moving chage foce at 90 to path speed v q magnetic field into plane of diagam The adius of cuvatue of the path of the beam mv =. qb Advancing Physics A 10

11 This is because the magnetic foce causes a centipetal acceleation v a =. Using F = ma gives mv qvb = mv and hence =. qb Note that witing the momentum mv as p, the elationship = p / B q emains coect even fo velocities appoaching that of light, when the momentum p becomes lage than the Newtonian value m v. The paticle acceleato ing of electomagnets acceleating electodes paticle beam in evacuated tube detectos In a paticle acceleato o collide, a ing of electomagnets is used to guide high-enegy chaged paticles on a closed cicula path. Acceleating electodes along the path of the beam incease the enegy of the paticles. The magnetic field stength of the electomagnets is inceased as the momentum of the paticles inceases, keeping the adius of cuvatue constant. Advancing Physics A 11

12 A TV o oscilloscope tube electon beam electon gun spot pictue lines taced out by the spot deflecting coils fluoescent sceen In a TV o oscilloscope tube, an electon beam is deflected by magnetic coils at the neck of the tube. One set of coils makes the spot move hoizontally and a diffeent set of coils makes it move vetically so it taces out a aste of descending hoizontal lines once fo each image. In a mass spectomete, a velocity selecto is used to ensue that all the paticles in the beam have the same speed. An electic field at ight angles to the beam povides a sideways deflecting foce q on each paticle. A magnetic field B (at ight angles to the electic field) is used to povide a sideways deflecting foce qvb in the opposite diection to that fom the electic field. The two foces ae equal and sum to zeo fo just the velocity v given by q = qvb, o v = / B. Only paticles with this velocity emain undeflected. Mass and enegy Mass and enegy ae linked togethe in the theoy of elativity. The theoy of elativity changes the meaning of mass, making the mass a pat of the total enegy of a system of objects. Fo example, the enegy of a photon can be used to ceate an electon-positon pai with mass 0.51 MeV / c each. Mass and momentum In classical Newtonian mechanics, the atio of two masses is the invese of the atio of the velocity changes each undegoes in any collision between the two. Mass is in this case elated to the difficulty of changing the motion of objects. Anothe way of saying the same thing is that the momentum of an object is p = m v. In the mechanics of the special theoy of elativity, the fundamental elation between momentum p, speed v and mass m is diffeent. It is: p = γmv with γ = 1 1 v / c At low speeds, with v << c, whee γ is appoximately equal to 1, this educes to the Newtonian value p = mv. negy The elationships between enegy, mass and speed also change. The quantity Advancing Physics A 1

13 total = γmc gives the total enegy of the moving object. This now includes enegy the paticle has at est (i.e. taveling with you), since when v = 0, γ = 1 and: est = mc This is the meaning of the famous equation = mc. The mass of an object (scaled by the facto c ) can be egaded as the est enegy of the object. If mass is measued in enegy units, the facto c is not needed. Fo example, the mass of an electon is close to 0.51 MeV. Kinetic enegy The total enegy is the sum of est enegy and kinetic enegy, so that: kinetic = total est This means that the kinetic enegy is given by: kinetic = ( γ 1) mc At low speeds, with v << c, it tuns out that γ - 1 is given to a good appoximation by: 1 ( γ 1) = ( v / c ) so that the kinetic enegy has the well-known Newtonian value: 1 kinetic = mv High enegy appoximations Paticle acceleatos such as the Lage Hadon Collide ae capable of acceleating paticles to a total enegy many thousands of times lage than thei est enegy. In this case, the high enegy appoximations to the elativistic equations become vey simple. At any enegy, since just the elativistic facto γ: γ = total est total = γmc and est = mc, the atio of total enegy to est enegy is This gives a vey simple way to find γ, and so the effect of time dilation, fo paticles in such an acceleato. Since the est enegy is only a vey small pat of the total enegy, kinetic total the elationship between enegy and momentum also becomes vey simple. Since v c momentum can be witten: p γmc and since the total enegy is given by total = γmc thei atio is simply: total c p, giving pc total This elationship is exactly tue fo photons o othe paticles of zeo est mass, which always tavel at speed c., the Advancing Physics A 13

14 Diffeences with Newtonian theoy The elativistic equations cove a wide ange of phenomena than the classical elationships do. Change of mass equivalent to the change in est enegy is significant in nuclea eactions whee extemely stong foces confine potons and neutons to the nucleus. Nuclea est enegy changes ae typically of the ode of MeV pe nucleon, about a million times lage than chemical enegy changes. The change of mass fo an enegy change of 1 MeV is theefoe compaable with the mass of an electon. Changes of mass associated with change in est enegy in chemical eactions o in gavitational changes nea the ath ae small and usually undetectable compaed with the masses of the paticles involved. Fo example, a 1 kg mass would need to gain 64 MJ of potential enegy to leave the ath completely. The coesponding change in mass is insignificant ( kg = 64 MJ / c ). A typical chemical eaction involves enegy change of the ode of an electon volt (= J). The mass change is about kg (= J / c ), much smalle than the mass of an electon. Appoximate and exact equations The table below shows the elativistic equations elating enegy, momentum, mass and speed. These ae valid at all speeds v. It also shows the appoximations which ae valid at low speeds v << c, at vey high speeds v c, and in the special case whee m = 0 and v = c. Conditions m > 0 v any value < c any massive paticle m > 0 v << c Newtonian m > 0 v c ultaelativistic Relativistic facto γ γ = γ = 1 1 v total est / c Total enegy Rest enegy Kinetic enegy Momentum total = γmc γ 1 total mc γ = total est total = γmc mc est = ( ) kinetic = γ 1 mc p = γmv est = mc 1 kinetic mv p mv est = mc kinetic p γmc total p c total m = 0 v = c γ is undefined = hf = 0 est = kinetic = total p = c photons Relativistic calculations of enegy and speed Calculating the speed of an acceleated paticle, given its kinetic enegy If the speed is much less than that of light, Newton s laws ae good appoximations. Newtonian calculation: Since the kinetic enegy K = (1/)mv, the speed v is given by: Advancing Physics A 14

15 v m K In an acceleato in which a paticle of chage q is acceleated though a potential diffeence V, the kinetic enegy is given by: K = qv, Thus: v qv m The Newtonian calculation seems to give an absolute speed, not a atio v/c. Relativistic calculation: A elativistic calculation mustn t give an absolute speed. It can only give the speed of the paticle as a faction of the speed of light. The total enegy total of the paticle has to be compaed with its est enegy mc. Fo an electon, the est enegy coesponding to a mass of kg is 0.51 MeV. A convenient elativistic expession is: total est = γ whee γ = 1 1 v c Since the total enegy total = mc qv, then: qv γ = 1 mc This expession gives a good way to see how fa the elativistic calculation will depat fom the Newtonian appoximation. The Newtonian calculation is satisfactoy only if γ is close to 1. The atio v / c can be calculated fom γ: v c = 1 1 γ. A ule of thumb As long as the acceleato enegy qv is much less than the est enegy, the facto γ is close to 1, and v is much less than c. The Newtonian equations ae then a good appoximation. To keep γ close to 1, say up to 1.1, the acceleato enegy qv must be less than 1/10 the est enegy. So fo electons, est enegy 0.51 MeV, acceleating potential diffeences up to about 50 kv give speeds faily close to the Newtonian appoximation. This is a handy ule of thumb. Acceleating voltage / kv Speed of electons (Newton) γ = 1 qv/mc mc = 0.51 MeV Speed of electons (instein) o in speed c c 1.5% c c 7% c c 14% Advancing Physics A 15

16 Acceleating voltage / kv Speed of electons (Newton) γ = 1 qv/mc mc = 0.51 MeV Speed of electons (instein) o in speed c c 6% c c >300% An example: a cosmic ay cosses the Galaxy in 30 seconds A poton of enegy 10 0 ev is the highest enegy cosmic ay paticle yet obseved (008). How long does such a poton take to coss the entie Milky Way galaxy, diamete of the ode 10 5 light yeas? The est enegy (mass) of a poton is about 1 GeV/c, o 10 9 ev/c. Then: total = γmc with γ = 1 1 v / c Inseting values: total = 10 0 ev/c and m = 10 9 ev/c gives: 10 γ = ev ev = The poton, tavelling at vey close to the speed of light, would take 10 5 yeas to coss the galaxy of diamete 10 5 light yeas. But to the poton, the time equied will be its wistwatch time τ whee: t = γτ τ = t γ 5 10 yea = s = 30 s The wistwatch time fo the poton to coss the whole Galaxy is half a minute. Fom its point of view, the diamete of the galaxy is shunk by a facto 10 11, to a mee 10 5 km. lecton volt The chage of the electon e = C. The chage e on the electon was measued in 1915 by Robet Millikan, who invented a method of measuing the chage on individual chaged oil doplets. Millikan discoveed that the chage on an oil doplet was always a whole numbe multiple of C. Physicists often measue the enegy of chaged paticles in the unit electon volt (ev). This is the wok done when an electon is moved though a potential diffeence of 1 volt. Since the chage of the electon is C, then 1 ev = J. The enegy needed to ionise an atom is of the ode of 10 ev. X-ays ae poduced when electons with enegy of the ode 10 kev o moe stike a taget. The enegy of paticles fom adioactive decay can be of the ode 1 MeV. Advancing Physics A 16

17 Summay Diagams Back to list of Contents The electic field between paallel plates Shapes of electic fields between two flat chaged plates Advancing Physics A 17

18 Two ways of descibing electic foces Two ways of descibing electical foces Action at a distance Action via electic field foces on chages fom combined attactions and epulsions by chages on plates epel attact attact epel chages on plates poduce electic field foces on chages fom electic field Foces act acoss empty space lectic field: foces act locally, field fills space Defining electic field field chage q foce F = F/q unit of is N C 1 Advancing Physics A 18

19 Field stength and potential gadient Potential gadients Contous and slopes UNTAIN AURANT Ski Tows SKI CNTR Aonach an Nid Slope is steep whee contous ae close. Diection of steepest slope is pependicula to contous. walk along contou to stay at same height slope = change in height h = distance x negative slope is downhill, deceasing height Advancing Physics A 19

20 Potential gadients Field and potential gadient 1000 V 1000 V 750 V 500 V 50 V 0 V 1000 V x slope = change in potential V = distance x V electic field = x dv o = if slope vaies continuously dx negative slope is downhill, deceasing potential Field stength = potential gadient Advancing Physics A 0

21 Field lines and equipotential sufaces Field lines and equipotentials A unifom field 1000 V electic field equipotential suface 1000V 750V 500V 50V field is at ight angles to equipotential suface quipotentials nea a lightning conducto 0V V V field lines V V no foce in these diections so potential is constant foce in this diection, so potential changes equipotentials nea conducto equipotential follows conducto suface Field lines ae always pependicula to equipotential sufaces Advancing Physics A 1

22 Invese squae law and flux Two ways of saying the same thing Invese squae law and flux of lines though a suface xpeiment Invese squae law Gauss idea flux of lines measue -field of a chage at q aea 4 diffeent distances q field test chage foce F 1 F expeimentally k = V C 1 m 1 F q k 4 Think of lines of as continuous. 0 F Numbe of lines though aea of q any sphee q 0 = C V 1 m 1 expeiments done by Coulomb (1780s) density of lines q xample: field between paallel plates 4 = density of lines no. of lines though aea A is A q aea A no. of lines = chage enclosed / chage on aea A is q = A chage density pe unit aea q q A = = A 0 0 The electic field can be consideed as the density of lines though a suface = 0 kq q 0 = constant chage enclosed numbe of lines = 0 Advancing Physics A

23 Radial fields in gavity and electicity Gavity and electicity an analogy Gavitational field and adius 0 0 lectic field and adius 0 0 g V gav = aea g g V elec = aea adius adius Gavity and electicity an analogy Gavitational potential and adius 0 0 lectic potential and adius 0 0 V gav V elec field g = slope = V gav / field = slope = V elec / V gav V elec adius adius Advancing Physics A 3

24 Gavity and electicity an analogy V gav = aea unde gaph of field g against field g = slope of V gav against g = GM V = GM foce between masses is attactive V elec = aea unde gaph of field against field = slope of V elec against = kq V elec = kq k = foce between like chages is epulsive Potential wells and hills a positive chage makes a potential hill fo positive chages but a potential well fo negative chages m a mass m makes a potential well fo othe masses Chages make potential enegy hills fo like chages and potential enegy wells fo unlike chages Advancing Physics A 4

25 How an electic field deflects an electon beam Deflections of electon beam by an electic field hot cathode electon gun deflection plates V anode spacing d foce electic field = vetical foce F = e = e V d V d acceleating potential diffeence zeo potential F = e V d hoizontal acceleation constant velocity vetical acceleation constant velocity A unifom electic field deflects a chaged paticle along a paabolic path Advancing Physics A 5

26 Foce, field, enegy and potential Relationships between foce, field, enegy and potential lecticity aea unde gaph of foce o field against Inteaction of two chages Foce (adial component) unit N F = q 1 q 4 0 Potential enegy unit J potential = q 1 q 4 0 divide by chage multiply by chage Behaviou of isolated chage Field (adial component) unit N C 1 q = 4 0 Potential unit V = J C 1 V electic = q 4 0 slope of gaph of potential enegy o potential against Relationships between foce, field, enegy and potential Gavity Inteaction of two masses Foce (adial component) unit N divide by mass Behaviou of isolated mass Field (adial component) unit N kg 1 aea unde gaph of foce o field against F = G m 1 m Potential enegy unit J potential = G m 1 m multiply by mass g = G m Potential (adial component) unit J kg 1 V gav = G m slope of gaph of potential enegy o potential against Advancing Physics A 6

27 Millikan s expeiment Disceteness of chage micoscope 5 mm light souce oil dop Chage on dop is changed by ionising ai in cell, using weak adioactive souce d F = q W = mg V When dop is held stationay: electic foce F = gavitational foce W F = q W = mg Unifom electic field = V d q = mg V = mgd q If chages q ae discete multiples n of electon chage e, then q = ne V = mgd ne Vn = constant Changes can be detected that ae due to single electons leading to a calculated value of e. Advancing Physics A 7

28 How a magnetic field deflects an electon beam Magnetic deflection positive ions beam velocity foce B-field negative chages e.g. electons foce on positive chage B v foce at ight angles to field and velocity of chage, F = qvb Magnetic fields deflect moving chaged paticles in cicula paths Advancing Physics A 8

29 Foce on a cuent: foce on a moving chage Foce on cuent: foce on moving chage lectic moto Moving chage B-field electic cuent I F L q F v Foce on cuent I in length L F = ILB cuent = I = chage flow time q t Foce on chage q at velocity v F = qvb IL = q L = qv t The foce which dives electic motos is the same as the foce that deflects moving chaged paticles Advancing Physics A 9

30 Measuing the momentum of moving chaged paticles Measuing the momentum of moving chaged paticles velocity v cicula path, adius foce F magnetic field B into sceen velocity v chage q motion in cicle magnetic foce m v = foce F foce F = qvb m v = qvb p = mv = qb at elativistic speed p = qb is still tue but p = mv Momentum of the paticle is popotional to the adius of cuvatue of its path Advancing Physics A 30

31 The ultimate speed Betozzi s demonstation The ultimate speed: Betozzi s demonstation acceleated electons tube detects electons passing 8.4 m dift space bunch of electons v aluminium plate detects electons aiving. Rise in tempeatue checks enegy time oscilloscope The esults: speed calculated 1 fom mv = qv speed of light The diffeence made by elativity As paticles ae acceleated speed v eaches a limit, c kinetic enegy K inceases without limit momentum p inceases without limit At all speeds K = qv 0 actual speed of electons 0 4 acceleating p.d./mv At low speeds p mv 1 K mv At high speeds v c K pc Poweful acceleatos can t incease the speed of paticles above c, but they go on inceasing thei enegy and momentum Advancing Physics A 31

32 Relativistic momentum p = γmv instein edefines momentum Poblem: time t depends on elative motion because of time dilation (chapte 1) Newton s definition of momentum p = mv p = m x t elativistic momentum p = mv instein s solution: Replace t by, the change in wistwatch time, which does not depend on elative motion both quantities identical at low speeds Newtonian momentum p = mv fom time dilation: = t = substitute fo x = v t 1 1v /c instein s new definition of momentum p = m p = m p = mv x x t v/c elativistic momentum p = mv Relativistic momentum p = mv inceases faste than Newtonian momentum mv as v inceases towads c Advancing Physics A 3

33 Relativistic enegy total = γmc instein ethinks enegy elativistic idea: Space and time ae elated. Teat vaiables x and ct elativistic momentum fo component of movement in space similaly. Being at est means moving in wistwatch time. p = m x so invent elativistic momentum fo movement in wistwatch time Just wite ct in place of x fom time dilation x = multiply p 0 by c, getting a quantity having units of enegy (momentum speed) elativistic momentum fo component of movement in time p 0 = m ct p 0 = mc p 0 = mc p 0 c = = mc t est = mc both cuves have same shape at low speeds total = mc Newtonian kinetic enegy 1 mv v/ c intepet enegy = mc 1 paticle at est: v = 0 and = 1 est = mc paticle has est enegy elativistic enegy paticle moving at speed v enegy mc geate than est enegy 3 at low speeds v << c kinetic enegy has same value as fo Newton kinetic = mc mc = ( 1)mc kinetic = ( 1)mc 1 ~ mv (see gaph) total = mc intepet as total enegy = kinetic enegy est enegy total = mc est = mc = total e st Total enegy = mc. Rest enegy = mc. Total enegy = kinetic enegy est enegy. Advancing Physics A 33

34 negy, momentum and mass negy, momentum and mass Key elationships 1 elativistic facto = 1v /c momentum p = mv total enegy total = mc est enegy est = mc kinetic = total est = total est low speeds enegy v << c ~ 1 high speeds v c >> 1 enegy kinetic enegy small compaed to total enegy total ~ est = mc 1 kinetic ~ mv lage est enegy nealy equal to total enegy total = mc kinetic = total est = ( 1)mc lage kinetic enegy nealy equal to total enegy est enegy est = mc est >> kinetic sam e es t enegy scaled down kinetic ~ total >> est est enegy est = mc est enegy small compaed to total enegy momentum momentum since = 1 momentum p = mv ~ mv since v ~ c momentum p = mv p ~ mv since total = mc p total c Kinetic enegy small compaed to est enegy Kinetic enegy lage compaed to est enegy Back to list of Contents Advancing Physics A 34