Magnetic Forces. Physics 231 Lecture 7-1

Transcription

1 Magnetic Foces Physics 231 Lectue 7-1

2 Magnetic Foces Chaged paticles expeience an electic foce when in an electic field egadless of whethe they ae moving o not moving Thee is anothe foce that chaged paticles can expeience even in the absence of an electic field but only when they ae motion A Magnetic Foce Magnetic Inteactions ae the esult of elative motion Physics 231 Lectue 7-2

3 Quick Note on Magnetic Fields Like the electic field, the magnetic field is a Vecto, having both diection and magnitude We denote the magnetic field with the symbol The unit fo the magnetic field is the tesla 1tesla = 1T = 1N / A m Thee is anothe unit that is also used and that is the gauss 4 1gauss = 10 T Unlike Electic Fields which begin and end on chages, Magnetic Fields have neithe a beginning no an end Physics 231 Lectue 7-3

4 Magnetic Foces Given a chage q moving with a velocity v in a magnetic field, it is found that thee is a foce on the chage This foce is popotional to the chage q popotional to the speed v pependicula to both v and popotional to sinφ whee φ is the angle between v and This can be summaized as F = qv This is the coss poduct of the velocity vecto of the chaged paticle and the magnetic field vecto Physics 231 Lectue 7-4

5 Right Hand Rule To get the esultant diection fo the foce do the following: 1. Point you index finge (and you middle finge) along the diection of motion of the chage v 2. Rotate you middle finge away fom you index finge by the angle θ between v and 3. Hold you thumb pependicula to the plane fomed by both you index finge and middle finge 4. You thumb will then point in the diection of the foce F if the chage q is positive 5. Fo q < 0, the diection of the foce is opposite you thumb Physics 231 Lectue 7-5

6 Magnetic Foces Thee is no foce if v and ae eithe paallel o antipaallel The foce is maximum when v and ae pependicula to each othe Sin(90) = 1 Sin(0) = Sin(180) = 0 x x x x x x x x x x x x v x x x x x x q F v q F = 0 v q F The foce on a negative chage is in the opposite diection Physics 231 Lectue 7-6

7 Example Thee points ae aanged in a unifom magnetic field. The magnetic field points into the sceen. 1) A positively chaged paticle is located at point A and is stationay. The diection of the magnetic foce on the paticle is: a) Right b) Left c) Into the sceen d) Out of the sceen e) Zeo The magnetic foce is given by F = qv ut v is zeo. Theefoe the foce is also zeo. Physics 231 Lectue 7-7

8 Example Thee points ae aanged in a unifom magnetic field. The magnetic field points into the sceen. 2) The positive chage moves fom point A towad. The diection of the magnetic foce on the paticle is: a) Right b) Left c) Into the sceen d) Out of the sceen e) Zeo The magnetic foce is given by F Physics 231 Lectue 7-8 = qv The coss poduct of the velocity with the magnetic field is to the left and since the chage is positive the foce is then to the left

9 Example Thee points ae aanged in a unifom magnetic field. The magnetic field points into the sceen. 3) The positive chage moves fom point A towad C. The diection of the magnetic foce on the paticle is: a) up and ight b) up and left c) down and ight d) down and left The magnetic foce is given by F Physics 231 Lectue 7-9 = qv The coss poduct of the velocity with the magnetic field is to the uppe left and since the chage is positive the foce is then to the uppe left

10 Motion due to a Magnetic Foce When a chaged paticle moves in a magnetic field it expeiences a foce that is pependicula to the velocity Since the foce is pependicula to the velocity, the chaged paticle expeiences an acceleation that is pependicula to the velocity The magnitude of the velocity does not change, but the diection of the velocity does poducing cicula motion The magnetic foce does no wok on the paticle Physics 231 Lectue 7-10

11 Motion due to a Magnetic Foce The magnetic foce poduces cicula motion with the centipetal acceleation being given by v 2 R whee R is the adius of the obit Using Newton s second law we have The adius of the obit is then given by The angula speed ω is given by ω = F = v = R qv R = q m 2 v = m R m v q Physics 231 Lectue 7-11

12 Motion due to a Magnetic Foce What is the motion like if the velocity is not pependicula to? We beak the velocity into components along the magnetic field and pependicula to the magnetic field The component of the velocity pependicula to the magnetic field will still poduce cicula motion The component of the velocity paallel to the field poduces no foce and this motion is unaffected The combination of these two motions esults in a helical type motion Physics 231 Lectue 7-12

13 Velocity Selecto An inteesting device can be built that uses both magnetic and electic fields that ae pependicula to each othe q v A chaged paticle enteing this device with a velocity will expeience both an electic foce F E and a magnetic foce = q E = qv F E v Physics 231 Lectue 7-13

14 Velocity Selecto q v F E F E If the paticle is positively chaged then the magnetic foce on the paticle will be downwads and the electic foce will be upwads If the velocity of the chaged paticle is just ight then the net foce on the chaged paticle will be zeo qv = qe v = E Physics 231 Lectue 7-14

15 Magnetic Foces We know that a single moving chage expeiences a foce when it moves in a magnetic field What is the net effect if we have multiple chages moving togethe, as a cuent in a wie? We stat with a wie of length l and coss section aea A in a magnetic field of stength with the chages having a dift velocity of v d The total numbe of chages in this section is then whee n is the chage density nal The foce on a single chage moving with dift velocity v d is given by Physics 231 Lectue 7-15 F So the total foce on this segment is = qv d F = nqv d Al

16 Magnetic Foce on a Cuent We have so fa that ut we also have that Caying Wie F = nqv d Al J = nqvd and I = J A Combining these, we then have that F = I The foce on the wie is elated to the cuent in the wie and the length of the wie in the magnetic field If the field and the wie ae not pependicula to each the full elationship is F I l The diection of l is the diection of the cuent = l Physics 231 Lectue 7-16

17 Cuent Loop in a Magnetic Field Suppose that instead of a cuent element, we have a closed loop in a magnetic field We ask what happens to this loop Physics 231 Lectue 7-17

18 Cuent Loop in a Magnetic Field Each segment expeiences a magnetic foce since thee is a cuent in each segment As with the velocity, it is only the component of the wie that is pependicula to that mattes Each of the two shote sides expeiences a foce given by F ' = I b cosφ in the diections shown Since the magnitudes ae the same, the net foce in the y-diection is F y = 0 No tanslational motion in the y-diection Physics 231 Lectue 7-18

19 Cuent Loop in a Magnetic Field Now fo the two longe sides of length a Each of these two sides expeiences a foce given by F = I a in the diections shown ut since the foces ae of the same magnitude but in opposite diections we have F x = 0 No tanslational motion in the x-diection Physics 231 Lectue 7-19

20 Cuent Loop in a Magnetic Field Thee is no tanslational motion in eithe the x- o y-diections While the two foces in the y-diection ae colinea, the two foces in the x-diection ae not Theefoe thee is a toque about the y-axis The leve am fo each foce is b sinφ 2 The net toque about the y-axis is Physics 231 Lectue 7-20 b τ = 2F sinφ = I a bsinφ 2

21 Cuent Loop in a Magnetic Field This toque is along the positive y-axis and is given by τ = I Asinφ The poduct IA is efeed to as the magnetic moment µ = I A We ewite the toque as τ = µ sinφ Physics 231 Lectue 7-21

22 Magnetic Moment We defined the magnetic moment to be µ = I It also is a vecto whose diection is given by the diection of the aea of the loop A The diection of the aea is defined by the sense of the cuent τ = µ We can now wite the toque as Physics 231 Lectue 7-22

23 Potential Enegy of a Cuent Loop As the loop otates because of the toque, the magnetic field does wok on the loop We can talk about the potential enegy of the loop and this potential enegy is given by U = µ The potential enegy is the least when µ and ae paallel and lagest when µ and ae antipaallel Physics 231 Lectue 7-23

24 Example Two cuent caying loops ae oiented in a unifom magnetic field. The loops ae nealy identical, except the diection of cuent is evesed. µ 1 µ 2 1) What diection is the toque on loop 1? a) clockwise b) counte-clockwise c) zeo The magnetic moment fo Loop 1, µ 1, points to the left, while that fo Loop 2, µ 2, points to the ight The toque is given by τ = µ ut since µ 1 and ae antipaallel, the coss poduct is zeo, theefoe the toque is zeo! Physics 231 Lectue 7-24

25 Example Two cuent caying loops ae oiented in a unifom magnetic field. The loops ae nealy identical, except the diection of cuent is evesed. µ 1 µ 2 2) How does the magnitude of the toques on the two loops compae? a) τ 1 > τ 2 b) τ 1 = τ 2 c) τ 1 < τ 2 Loop 1: Since µ 1 points to the left the angle between µ 1 and is equal to 180º theefoe τ 1 = 0. Loop 2: Since µ 2 points to the ight the angle between µ 2 and is equal to 0º theefoe τ 2 = 0. So the two toques ae equal! Physics 231 Lectue 7-25

26 Example Two cuent caying loops ae oiented in a unifom magnetic field. The loops ae nealy identical, except the diection of cuent is evesed. µ 1 µ 2 3) Which loop occupies a potential enegy minimum, and is theefoe stable? a) Loop 1 b) Loop 2 c) the same The potential enegy is given by U µ Fo Loop 1 the potential enegy is then U 1 = +µ 1 While fo Loop 2 the potential enegy is then U 2 = µ The potential enegy fo Loop 2 is less than that fo Loop 1 = Physics 231 Lectue 7-26

27 Motion of Cuent Loop The cuent loop in its motion will oscillate about the point of minimum potential enegy If the loop stats fom the point of minimum potential enegy and is then displaced slightly fom its position, it will etun, i.e. it will oscillate about this point This initial point is a point of Stable equilibium If the loop stats fom the point of maximum potential enegy and is then displaced, it will not etun, but will then oscillate about the point of minimum potential enegy This initial point is a point of Unstable equilibium Physics 231 Lectue 7-27

28 Moe Than One Loop If the cuent element has moe than one loop, all that is necessay is to multiply the pevious esults by the numbe of loops that ae in the cuent element Physics 231 Lectue 7-28

29 Hall Effect Thee is anothe effect that occus when a wie caying a cuent is immesed in a magnetic field Assume that it is the positive chages that ae in motion These positive chages will expeience a foce that will cause them to also move in the diection of the foce towads the edge of the conducto, leaving an appaent negative chage at the opposite edge Physics 231 Lectue 7-29

30 Hall Effect The fact that the thee is an appaent chage sepaation poduces an electic field acoss the conducto Eventually the electic field will be stong enough so that subsequent chages feel an equivalent foce in the opposite diection q Ee = qvd o Ee = vd Since thee is an electic field, thee is a potential diffeence acoss the conducto which is given by V = Ee d = v d d Physics 231 Lectue 7-30

31 Hall Effect The Hall Effect allows us to detemine the sign of the chages that actually make up the cuent If the positive chages in fact constitute the cuent, then potential will be highe at the uppe edge If the negative chages in fact constitute the cuent, then potential will be highe at the lowe edge Expeiment shows that the second case is tue The chage caies ae in fact the negative electons Physics 231 Lectue 7-31