Mon., 3/9 Tues., 3/10 Wed., 3/11 Thurs., 3/12 Fri., 3/ 13. RE19 HW19:RQ.42, 49, 52; P.61, 66, 69 RE20, Exp new RE ,3-4 Magnetic Force

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1 Mon., 3/9 Tues., 3/10 Wed., 3/11 Thus., 3/12 Fi., 3/ 13 Mon., 3/16 Tues., 3/17 Wed., 3/18 Thus., 3/19 Fi., 3/ ,3-4 Magnetic Foce 20.2,5 Cuent and Motional Emf Quiz Ch 19, Lab 8 Cycloton & Electon Mass Lab Refeence Fames and Relatiity, Toque Bonus: Phys. S. Thesis 4pm Dipole s Potential Enegy, Motos & Geneatos Reiew Exam 2 (17-20) Magnetic Field and Moing Chages E Flux and Gauss s Law RE19 HW19:RQ.42, 49, 52; P.61, 66, 69 RE20, Exp new RE21 RE22, Exp 26 HW20:RQ.28, 34, 36; P.45, 56, Lab 7 RE23, Lab Notebook Load VPython Check Missing a few people s homewok Who s planning on coming to the S. Pesentations? Magnetic Foce on a Moing Chage: Back to Magnetism. Now that we hae a good feel fo cuents and cicuits, we e eady to get back to that inteaction that elies on cuents / moing chages: magnetism. Recall that when we fist stated talking about Magnetism, I bought in a plank with two wies and we saw that when anti-paallel cuents an though them, they epelled. I suggested that paallel cuents attacted, And this, I said, was the heat of magnetism. I also flashed up a athe ugly expession fo the magnetic foce between two moing chages: F mag 2 1 µ o 4π q ( q ) 22 ˆ (neglecting tems on ode of (/c) 2 ) Paticulaly consideing all that those two coss poducts entail, this is a athe ugly expession. So, aside fom its othe meits, defining a magnetic field, µ o ( q11 ˆ 2 1 ) Bmag ( 1 2 ) 2 4π 2 1 allowed us to conceptually and mathematically beak up this inteaction and teat it piecewise What field does moing chage 1 ceate at location 2, and then what foce does that exet on chage 2. We d spent some time mapping out the magnetic fields due to moing chages. Now it s time to look at the esulting foces.

2 The foce on a moing chaged paticle in a magnetic field is: F = q B ) mag mag ( 1 2 That coss-poduct means that the foce is pependicula to both the elocity and the magnetic field (use RHR to find the diection). The size of the magnetic foce is: F mag = F mag = qbsin θ, whee θ is the angle between the elocity and magnetic field ectos. As you ae quite familia by now, don t foget that the sign of the chage effects the diection. The combined electic and magnetic foces o the Loentz foce is: F = qe + q B Magnetic Foce does not change a paticle s speed (just diection of motion): Gien all the coss-poducting, one unique thing about the magnetic foce (and theefoe the esulting acceleation) is that, since it s always pependicula to a paticle s elocity because B is pependicula to both and B, it alone can t effect the speed, just the diection. VPython: 20_helix_in_B_Piate (Case = 1) Wok and Enegy. o That means that it can t effect the kinetic enegy (which depends on speed), and, fo that matte, it can t do wok. o In a ey shot time, the paticle will undego a displacement d l in the diection of its elocity. Theefoe, the wok done on the paticle in that time is dw = F mag d l which is zeo. That means that the magnetic foce cannot change the paticle s kinetic enegy o speed. The magnetic foce can change the diection of the paticles motion and its elocity. Execises: (fom Ex ) Daw the net foce on the paticle in each of the figues Julie s 1 st two slides (1) (2) Monday, Mach. 9th,

3 Answes: (3) (4) (1) (2) (3) (4) Julie s 4 th slide Example: Velocity Selecto Suppose a poton is moing elatie to electic and magnetic fields as shown below. The electic and magnetic foces ae in opposite diections, so they will cancel if they ae the same size. This equies: ee = ebsin 90 o E = B If a poton is taeling at a diffeent speed it will be deflected. Monday, Mach. 9th,

4 Cicula motion: Suppose a chaged paticle is moing pependicula to a unifom magnetic field. The foce will be pependicula to the elocity, so it will change the paticles diection but not its speed. Theefoe, the sized of the foce will not change eithe. The paticle will tael in a cicle. VPython 20_helix_in_B The moment pinciple (Newton s 2 nd law) is: dp Fo cicula motion, the change in momentum is dpˆ p dt dt =. F net dθ = p = p ω = F dt net But the net foce is the magnetic foce which is ω = 2 π = T p R = R F net F = F = qbsin θ = qb mag mag (fo 90 ). Then, p = q B so R Application: Paticle Identification p = qrb Field Tip: Omega pictue. Knowing the stength of the field and magnitude of chage, usually e, the adius of the path and its bend (clockwise s. counteclockwise) tells the paticle s momentum and sign of chage. The chaged paticle moes in a cicle of adius: p R =. q B The time T (the peiod) to go aound a cicle (at constant speed, since magnetic foces don t change the speed) is found using: so the peiod is: T = 2πR = 2πR T, = 2πm. (assuming non-elatiistic speeds) q B Monday, Mach. 9th,

5 Note that the time to go aound a cicle doesn t depend on the speed, because a faste paticle will make a lage cicle (fo the same magnetic field). Application: the Cycloton We can take adantage of this cicula motion and constant peiod to acceleate paticles to high enegies. In the aeas of Paticle and Nuclea Physics, one of the best ways we hae to expeimentally pobe nuclei and poduce exotic paticles is with ey high-enegy paticle collisions. Fo example, you could collide high enegy potons. As you may ecall fom scatteing poblems fom last semeste, the faste thei going / the highe thei initial kinetic enegies, the close they can get to each othe befoe deflecting. Now, fo acceleating chaged paticles just a little, it s easy enough to use big capacitos inset the paticle between two plates, and it gets swept up by the field. Howee, if you want to get up to kinetic enegies of, say 25 MeV, that would be one huge capacito. Hee s an elegant altenatie that could use much, much lowe acceleating oltages. As we e just seen, a chaged paticle will cicle aound a constant magnetic field, and the peiod of that 2πm obit is, fo <<c, independent of the speed. In paticula, T =. Say you hae an 0.1 Tesla q B kg 7 magnetic field, then a poton would cycle at T = π = s. A cycloton C 0.1T takes adantage of this to acceleate a steam of paticles to faily high elocities. A adioactie souce that emits potons is placed in the gap between two hollowed out dees as shown. They ae wied to an oscillating oltage souce so that the oltage diffeence VL R oscillates between positie and negatie. On top of that, the dees ae sandwhiched between two magnets, say, a S pole just aboe and a N pole just below so a faily unifom magnetic field points up though the appaatus. So, what happens. When a poton is emitted in the middle of the gap, it s swept off to the left o ight accoding to the diection of the electic field at that instant, let s say left. When it entes the left dee, it s in a egion of nealy zeo electic field, but it still feels the magnetic field, so it 7 acs. Half a peiod late, ( T / 2 = s, in this paticula example), the poton cosses back into the gap, this time headed ight. Now, if, in the meantime, the oltage diffeence between the two dees has been flipped so the electic field now points ight, so if it oscillates with peiod T/2, then the poton gets acceleated some moe, until it entes the ight dee. This can go on, and on, with the poton spialing out futhe and futhe. Monday, Mach. 9th,

6 One beautiful thing is, though the poton is getting acceleated / its speed is gowing, so is its adius just so that it maintains a constant peiod. So the oltage diffeence of the two dees just needs to oscillate with a constant peiod of T/2 to, each time the poton entes the gap, poide the coect electic field to acceleate it some moe. Anothe consequence of this constancy is that a whole steam of potons, all at diffeent speeds / adii can be acceleated they all hae the same obital peiod. Now, athe than needing one 25 MV capacito, we need, say, a 2.5 kv gap, though which the poton passes 100 times. VPython: Cycloton Deflection of Chaged paticles by Eath s magnetic field: Chaged paticles fom the Sun ae deflected by the Eath s magnetic field. Positie paticles tend to spial along the magnetic field and negatie paticles spial the othe way. To see this, you hae to look at what happens afte the initial deflection. The positie paticles ae initially deflected out of the page, then somewhat upwad. These paticles stiking the atmosphee nea the poles case the Auoa Boealis (nothen lights) and Auoa Austalialis (southen lights). Monday, Mach. 9th,

7 The Hall effect: Julie s last slide Q20.4a Diection of E inside ba? x x +y y +z z none (zeo magnitude) Q20.4b If mobile chages ae negatie, diection of motion inside ba? 1. +x 2. x 3. +y 4. y 5. +z 6. z 7. none (zeo magnitude) Q20.4c Diection of Fmag on moing negatie chage? 1. +x 2. x 3. +y 4. y 5. +z 6. z 7. none (zeo magnitude) Q20.4d Monday, Mach. 9th,

8 If mobile chages ae negatie, sign of oltmete eading? (Voltmete eads positie if + lead is connected to highe potential location) 1. positie 2. negatie 3. zeo Q20.4e If mobile chages ae positie, diection of motion inside ba? 1. +x 2. x 3. +y 4. y 5. +z 6. z 7. none (zeo magnitude) Q20.4f If mobile chages ae positie, diection of magnetic foce? 1. +x 2. x 3. +y 4. y 5. +z 6. z 7. none (zeo magnitude) Monday, Mach. 9th,

9 Q20.4g If mobile chages ae positie, sign of oltmete eading? (Voltmete eads positie if + lead is connected to highe potential location) 1. positie 2. negatie 3. zeo Clicke Questions 20.4a-g The magnetic foce can be used to detemine the sign of the chage caies in a conducto. Typically, a wide conducting stip is used. A magnetic field is applied pependicula to the diection of the cuent (and electic field in the metal). Suppose negatie electons ae moing in the metal. They would be deflected downwad in the diagam below. The bottom of the stip will become negatiely chaged and the top positiely chaged. Eentually, the tansese electic field E due so the chages on top and bottom will esult in an electic foce that will balance the magnetic foce Eentually, the tansese electic field E due so the chages on top and bottom will esult in an electic foce that will balance the magnetic foce. This occus when: ee = e B, which means that the size of the potential diffeence between the top and bottom is: V = E h, whee h is the height. This can be measued with a oltmete. Fo a mete connected as shown aboe, the mete will gie a positie eading because the potential at the + socket is highe. Suppose positie potons wee moing in the opposite diection so that the conentional cuent was the same as aboe. How would the situation change? Monday, Mach. 9th,

10 The diection of the magnetic foce on the potons would also be downwad because both the sign of the chage and the diection of the elocity ae switched. The top and bottom sufaces would become chaged in the opposite way. The condition fo the tansese electic field would be the same, so the size of the potential diffeence would be the same. Howee, a oltmete connected in the same way will gie a negatie eading because the potential is lowe at the + socket. It is the sign of the potential diffeence that tells what the sign of the chage caies is. The chage caie density can also be detemined: I V = E h = Bh = Bh n = q na I Bh q VA Fo next time: emind myself about Quantum and Factional Quantum Hall Effects. DEMO: Bing the Hall effect appaatus to class! Monday, Mach. 9th,

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