PHYS 55 Pactice Test Solutions Fall 8 Q: [] poton an an electon attact each othe electicall so, when elease fom est, the will acceleate towa each othe Which paticle will have a lage acceleation? (Neglect gavitational attaction) a) The acceleations ae the same since the Coulomb foces ae equal b) The electon since it is lighte c) The poton since it is heavie ) The paticles won t acceleate since the foces cancel each othe out Solution: b The chages act upon each othe with foces which in the spiit of Newton s 3 law fom action-eaction pais So, the poton an the electon will attact each othe with foces with same magnitue Hence, in confomit with Newton s n law, thei acceleations will epen on thei elative masses: a m F F m a m a a a e p e e p p e p ap me That is, the acceleation of the electon is lage than the acceleation of the poton b the same facto the electon is lighte than the poton Q: [] Two chages ae epesente in the figue with some electic fiel lines ase on the patten, which chage is lage an b how much? a) The positive chage, thee times b) The negative chage, thee times c) The chages ae equal ) The positive chage, two times The numbe of electic fiel lines is popotional to the intensit of the electic fiel so the positive chage is lage since it has moe lines suouning it lso, the fiel of a chage n times lage than anothe chage shoul be epesente b n times moe fiel lines So, since in ou case thee ae lines eiting the positive chage but onl of them en on the negative chage, the positive chage must be 3 times lage Q3: [] poton entes with hoizontal velocit v in the vacuume space between the hoizontal plates of a chage paallel-plate capacito Which of the fou paths epicte on the iagam will be most likel followe b the poton? (staight line) (ac of cicle) (ellipse) (paabola) of the above C v The electic fiel between the plates is unifom, so the foce acting on the poton is constant, so its acceleation is constant Since the poton caies a positive chage, the foce is in the iection of the fiel, veticall upwa in the positive -iection Consequentl, onl the -component of the paticle s motion is acceleate wheeas the -component is not Hence, the time epenence of the cooinates is given b F e at t E t m m e v t an (las, PHY 7 bites us fom behin ) Substituting the time between the two equations, we get a quaatic epenenc between an, that is, the path is paabolic: e ee E m v mv
PHYS 55 Pactice Test Solutions Fall 8 Q: [] n electon is place in the inteio of an imagina sphee one thi the aius fom the ight hemispheic suface Then the electic flu though the suface of the sphee is a) e b) 3 c) ) e 9 The sphee can be chosen as a Gaussian Then, b Gauss s law, the flu though its close suface is popotional to the chage within iespective of its position: e -e Q5: [] wie but ve thin metallic plate of thickness hols a positive chage E E Which of the ajacent cuves most likel epesents the magnitue of electic fiel along an -ais pepenicula though the mile of the plate? E C E The electic fiel in the inteio of a metal must be zeo, so none of the cuves with a finite fiel insie ( an C) can chaacteize it On the othe han, since the plate is much wie than the egion along -ais whee the fiel is measue, it can be appoimate as two infinite sheets of positive chage poucing constant fiels in the vicinit This conition is satisfie b, but not b Q6: [] Which of the ajacent cuves most likel epesents the electic potential of a chage metallic sphee in electostatic equilibium as a function of the istance fom the cente of the sphee R is the aius of the sphee Thee is no net chage insie a metal, so the inne fiel is zeo Theefoe, the potential must be constant thoughout the inteio of the conucto an it thee is no iscontinuit at the suface as on gaph R R C R R Q7: [] Thee point chages with the same magnitue q = 6 μc ae locate in the cones of an equilateal tiangle of sie = 56 mm What is the wok one on the positive chage to bing it to its cone fom infinit? a) 89 J b) 56 J c) 8 J ) J of the above: insufficient infomation The wok one b the two negative chages to bing the positive chage fom infinit to the cone of the tiangle is given b the iffeence of potential between the two positions multiplie b the tanspote chage The positive chage moves though the potential ceate b the two negative point chages Theefoe we get q q q W q q q 8 J -q -q q
PHYS 55 Pactice Test Solutions Fall 8 Q8: [] The figue epesents the electic potential of a ipole positive test chage is elease in vicinit with an initial velocit oiente as in the figue Which of the inicate tajectoies will be most likel followe b the test chage? of the above The positive test chage spontaneousl moves in the iection of the electic fiel pointing own the gaient of potential towa egions with lowe electic potentials, so it will ten to fall into the potential well coesponing to the negative chage of the ipole Since the test chage has an initial velocit, it will fall own into the sink along a spial tajecto much like the motion of a mass own a whilpool C v Q9: [] oh s moel fo the hogen atom suggeste that the electon moves along cicula obits about the nucleus Which of the ajacent maps most likel epesents the electic potential eneg of the electon in the fiel of the poton? Solution: b The poton ceates a positive potential (a peak) which tens asmptoticall to infinit in its poimit ut the electon has negative chage, so it is evolving on the wall of a potential eneg well since e infinite peak C equipotential e e e - - - infinite well finite peak e - Q: [] The electic fiel in the vicinit of a continuous istibution of chage with aial smmet vaies with the istance fom the cente as / 3 How oes the electic potential epen on? a) / b) ln c) / ) / e) The electic potential is constant Solution: c The fiel is elate to the gaient of potential In this aiall smmetic case, we have E E ~ Notice that the situation is phsicall possible if the chage is non-unifoml istibute 3
PHYS 55 Pactice Test Solutions Fall 8 P: Thee point chages Q = 6 μc an Q = 3 μc ae fie in the cones of a vetical equilateal tiangle of sie = cm, as in the figue test chage q = 6 μc of mass m is place in point whee it floats in static equilibium at height h = / cm above the chages Q a) [] se vecto aows to epesent the electic fiels E, E, an E 3 ceate b the thee chages in point The fiel eists in point inepenent of the pesence of the test chage The souce chages ae positive, such that the espective fiels ae aows pointing aiall awa fom each chage b) [7] Split the vectos in an -components an fin a smbolical epession fo the magnitue of the net electic Q Q α = 5 fiel in point It shoul epen on Q, Q, an constants Then epesent the vecto net fiel on the figue / / an calculate the magnitue numeicall Note that the -components of the fiels ceate b the bottom chages cancel each othe out, so we emain onl with thei -components which ae equal Theefoe, the net fiel is given b Q Q E Esin E k k, whee an ae the istances between the espective chages an point Fom the geomet of the aangement, Theefoe 3, h 3 Q Q k Q E k k Q 3 3 The negative sign means that the net fiel is oiente ownwa 7 m c) [3] Repesent the electic foce acting on chage q Then use equilibium to calculate the mass m of the test chage The test chage is negative such that the foce will be veticall upwa (against the fiel) Since the chage is in equilibium F mg qe mg m q E g 39 kg e ) [] Suppose that the uppe chage Q is emove Calculate the iffeence of electic potential ceate b the emaining chages Q between points an oth chages contibute to the potentials equall Theefoe Q Q Q 5 k k k 8 e) [] The test chage q is elease fom est an moves ownwa Calculate its spee in point Since the test chage falls fom est its balance of final an initial potential enegies gives us the final kinetic eneg mg q q mv mv mg q 3 J, such that the spee is v 3 39 m s m s Q / q α
PHYS 55 Pactice Test Solutions Fall 8 P: clinical wie of aius a is electicall chage with unifom linea ensit λ = 3 - C/m Consie the cline infinitel long an aligne with -ais, as in the figue The questions below will lea ou though the most common calculation of the electic fiel an potential in point P(,), at a istance = cm fom the ais of the cline a) [] Sketch a clinical Gaussian of aius an abita height L aoun the wie epesente on the figue enote q the chage enclose b this suface an wite it own in tems of λ an L q L b) [5] se Gauss s Law to povie a smbolical epession fo the magnitue of the electic fiel E in point P in tems of, λ an ε Then calculate it numeicall Inicate on the figue the iection of the vecto E in point P Since the electic fiel must espect the aial smmet of the chage, the electic flu though the Gaussian must be nonzeo onl though the cuve suface (so no flu though the top an bottom) Moeove, b smmet, the magnitue of the fiel must be constant on these cuve sufaces Consequentl, b Gauss s Law, we get E E L q E L E b Gauss's Law Gaussian L E Since the chage on the wie is negative, the fiel vecto points aiall insie as inicate b the epession fo the component E which is negative Numeicall, the fiel magnitue is E 8 m c) [5] se the epession fo the electic fiel to fin a smbolical epession fo the iffeence of electic potential between point P an a point outsie the wie at istance fom P, locate on the -ais Fo an consevative fiel with aial smmet, we have the elationship between the fiel stength an the potential: E ˆ ; Hence, we get E E ln, whee the pime is a customa wa to mak a umm vaiable since is a limit of integation The goun o suface of zeo potential can be taken on the suface of the wie, = a, an then this is an epession fo the potential at istance ) [6] Now suppose that a poton is launche in point P moving to the ight along -ais with initial spee v = 5 m/s What istance oes the poton tavel befoe it stops? We can use consevation of eneg Since thee is no nonconsevative foce acting on the poton, the consevation of eneg between the initial configuation (point P) an the point at position whee it stops (since the poton is attacte b the negative chage on the wie) is given b: mv e e Hence, we can calculate the istance = tavele b the poton e ln mv e mv mv mv e e ln e e 5 mm e L a P (,) 5