nearingzero (nz182.jpg)

1. http://www.nearingzero.net (nz182.jpg)

2. Today’s agenda: Induced emf. You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. Faraday’s Law. You must be able to use Faraday’s Law to calculate the emf induced in a circuit. Lenz’s Law. You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf. Generators. You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators. Back emf. You must be able to use Lenz’s law to explain back emf.

3. Induced emf and Faraday’s Law Magnetic Induction We have found that an electric current can give rise to a magnetic field… I wonder if a magnetic field can somehow give rise to an electric current…

4. It is observed experimentally that changes in magnetic flux induce an emf in a conductor. B An electric current is induced if there is a closed circuit (e.g., loop of wire) in the changing magnetic flux. I B A constant magnetic flux does not induce an emf—it takes a changing magnetic flux.

5. N S v move magnet toward coil Note that “change” may or may not not require observable (to you) motion.  A magnet may move through a loop of wire, or a loop of wire may be moved through a magnetic field. These involve observable motion.  A magnet may move through a loop of wire B I  I region of magnetic field this part of the loop is closest to your eyes change area of loop inside magnetic field N S rotate coil in magnetic field

6. changing I induced I changing B  A changing current in a loop of wire gives rise to a changing magnetic field (predicted by Ampere’s law)  A changing current in a loop of wire gives rise to a changing magnetic field (predicted by Ampere’s law) which can induce a current in another nearby loop of wire.  A changing current in a loop of wire In the this case, nothing observable (to your eye) is moving, although, of course microscopically, electrons are in motion. Induced emf is produced by a changing magnetic flux.

7. Today’s agenda: Induced emf. You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. Faraday’s Law. You must be able to use Faraday’s Law to calculate the emf induced in a circuit. Lenz’s Law. You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf. Generators. You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators. Back emf. You must be able to use Lenz’s law to explain back emf.

8. We can quantify the induced emf described qualitatively in the last few slides by using magnetic flux. Experimentally, if the flux through N loops of wire changes by dB in a time dt, the induced emf is Faraday’s Law of Magnetic Induction Faraday’s law of induction is one of the fundamental laws of electricity and magnetism. I wonder why the – sign… Your text, page 301, shows how to determine the direction of the induced emf. Argh! Lenz’s Law, coming soon, is much easier.

9. Faraday’s Law of Magnetic Induction In the equation is the magnetic flux. This is sometimes shown as another expression of Faraday’s Law: We’ll use this version in a “later” lecture.

10. N S I v + - Example: move a magnet towards a coil of wire. N=5 turns A=0.002 m2

11. Possible homework hint: if B varies but loop  B. Ways to induce an emf:  change B  change the area of the loop in the field Possible homework hint: for a circular loop, C=2R, so A=r2=(C/2)2=C2/4, so you can express d(BA)/dt in terms of dC/dt.

12.  changing current changes B through conducting loop a I B b Possible Homework Hint. The magnetic field is not uniform through the loop, so you can’t use BA to calculate the flux. Take an infinitesimally thin strip. Then the flux is d = BdAstrip. Integrate from a to b to get the flux through the strip.

13. =90 =45 =0 Ways to induce an emf (continued):  change the orientation of the loop in the field

14. Today’s agenda: Induced emf. You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. Faraday’s Law. You must be able to use Faraday’s Law to calculate the emf induced in a circuit. Lenz’s Law. You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf. Generators. You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators. Back emf. You must be able to use Lenz’s law to explain back emf.

15. N S Experimentally… Lenz’s law—An induced emf always gives rise to a current whose magnetic field opposes the change in flux.* I v + - If Lenz’s law were not true—if there were a + sign in Faraday’s law—then a changing magnetic field would produce a current, which would further increase the magnetic field, further increasing the current, making the magnetic field still bigger… *Think of the current resulting from the induced emf as “trying” to maintain the status quo—to prevent change.

16. …violating conservation of energy and ripping apart the very fabric of the universe…

17. No “Quiz” time (no time for real or practice quiz today!)

18. Possible Demos: induced emf. Then skip to slide 23.

19. Practice with Lenz’s Law. In which direction is the current induced in the coil for each situation shown? (counterclockwise) (no current)

20. (counterclockwise) (clockwise)

21. Rotating the coil about the vertical diameter by pulling the left side toward the reader and pushing the right side away from the reader in a magnetic field that points from right to left in the plane of the page. (counterclockwise)

22. Faraday’s Law You can use Faraday’s Law to calculate the magnitude of the emf (or whatever the problem wants). Then use Lenz’s Law to figure out the direction of the induced current (or the direction of whatever the problem wants). The direction of the induced emf is in the direction of the current that flows in response to the flux change. We usually ask you to calculate the magnitude of the induced emf ( || ) and separately specify its direction. Magnetic flux is not a vector. Like electrical current, it is a scalar. Just as we talk about current direction (even though it is not a vector), we often talk about flux direction (even though it is not a vector). Keep this in mind if your recitation instructor talks about the direction of magnetic flux.

23. Today’s agenda: Induced emf. You must understand how changing magnetic flux can induce an emf, and be able to determine the direction of the induced emf. Faraday’s Law. You must be able to use Faraday’s Law to calculate the emf induced in a circuit. Lenz’s Law. You must be able to use Lenz’s Law to determine the direction induced current, and therefore induced emf. Generators. You must understand how generators work, and use Faraday’s Law to calculate numerical values of parameters associated with generators. Back emf. You must be able to use Lenz’s law to explain back emf.

24. Motional emf: an overview An emf is induced in a conductor moving in a magnetic field. Your text introduces four ways of producing motional emf. 1. Flux change through a conducting loop produces an emf:rotating loop. B start with this A  side view derive these

25. 2. Flux change through a conducting loop produces an emf: 2. Flux change through a conducting loop produces an emf:expanding loop. v B B start with these                                     ℓ                         dA x=vdt derive these

26. 3. Conductor moving in a magnetic field experiences an emf: 3. Conductor moving in a magnetic field experiences an emf: magnetic force on charged particles. start with these v B B             –                         ℓ                         + derive this You could also solve this using Faraday’s Law by constructing a “virtual” circuit using “virtual” conductors.

27.                                                                                                     4. Flux change through a conducting loop produces an emf:moving loop. start with this derive these

28. Let’s look in detail at each of these four ways of using motion to produce an emf. Method 1…

29. B A  side view S N Generators and Motors: a basic introduction Take a loop of wire in a magnetic field and rotate it with an angular speed . Choose 0=0. Then Generators are an application of motional emf.

30. B A  side view If there are N loops in the coil The NBA equation! || is maximum when  = t = 90° or 270°; i.e., when B is zero. The rate at which the magnetic flux is changing is then maximum. On the other hand,  is zero when the magnetic flux is maximum.

31. emf, current and power from a generator None of these are on your starting equation sheet!

32. Example: the armature of a 60 Hz ac generator rotates in a 0.15 T magnetic field. If the area of the coil is 2x10-2 m2, how many loops must the coil contain if the peak output is to be max = 170 V? “Legal” for me, because I just derived it (but not for you)! I will not work this in lecture. Please study it on your own! Know where the 2 comes from.

33. Let’s look in detail at each of these four ways of using motion to produce an emf. Method 2…

34. B B                                                             Another Kind of Generator: A Slidewire Generator Recall that one of the ways to induce an emf is to change the area of the loop in the magnetic field. Let’s see how this works. v A U-shaped conductor and a moveable conducting rod are placed in a magnetic field, as shown. ℓ The rod moves to the right with a constant speed v for a time dt. dA vdt x The rod moves a distance vdt and the area of the loop inside the magnetic field increases by an amount dA = ℓ vdt .

35. The loop is perpendicular to the magnetic field, so the magnetic flux through the loop is B = = BA. The emf induced in the conductor can be calculated using Faraday’s law: v B B                         ℓ                         dA             vdt x B and v are vector magnitudes, so they are always +. Wire length is always +. You use Lenz’s law to get the direction of the current.

36. B                               Direction of current? The induced emf causes current to flow in the loop. Magnetic flux inside the loop increases (more area). v System “wants” to make the flux stay the same, so the current gives rise to a field inside the loop into the plane of the paper (to counteract the “extra” flux). ℓ I dA vdt x Clockwise current! What would happen if the bar were moved to the left?

37. As the bar moves through the magnetic field, it “feels” a force v B       A constant pulling force, equal in magnitude and opposite in direction, must be applied to keep the bar moving with a constant velocity. FM FP             ℓ             I x

38. Power and current. If the loop has resistance R, the current is v B       And the power is             ℓ             I (as expected). x Mechanical energy (from the pulling force) has been converted into electrical energy, and the electrical energy is then dissipated by the resistance of the wire.

39. You might find it useful to look at Dr. Waddill’s lecture on Faraday’s Law, from several semesters back. Click here to view the lecture. If the above link doesn’t work, try copying and pasting this into your browser address bar: http://campus.mst.edu/physics/courses/24/Handouts/Lec_18.ppt

40. Let’s look in detail at each of these four ways of using motion to produce an emf. Method 3…

41. Example 3 of motional emf: moving conductor in B field. Motional emf is the emf induced in a conductor moving in a magnetic field. “up” v B B If a conductor (purple bar) moves with speed v in a magnetic field, the electrons in the bar experience a force             –                         ℓ                         + The force on the electrons is “up,” so the “top” end of the bar acquires a net – charge and the “bottom” end of the bar acquires a net + charge. The charges in the bar are “separated.” This is a simplified explanation but it gives you the right “feel.”

42. The separated charges in the bar produce an electric field pointing “up” the bar. The emf across the length of the bar is “up” The electric field exerts a “downward” force on the electrons: v B B             –                         ℓ An equilibrium condition is reached, where the magnetic and electric forces are equal in magnitude and opposite in direction.                         + Skip to slide 44.

43. Homework Hint! When the moving conductor is “tilted” relative to its direction of motion (when is not perpen-dicular to the conductor), you must use the “effective length” of the conductor. v B B             –                          ℓeffective             +             ℓeffective= ℓ sin  if you use the if you define  as the angle relative to the horizontal Caution: you do not have to use this technique for homework problems 9.23 (assigned this semester) or 9.26. Be sure to understand why!

44. Let’s look in detail at each of these four ways of using motion to produce an emf. Method 4…

45. Example 4 of motional emf: flux change through conducting loop. (Entire loop is moving.) I’ll include some numbers with this example. Remember, it’s the flux change that produces the emf. Flux has no direction associated with it. However, the presence of flux is due to the presence of a magnetic field, which does have a direction, and allows us to use Lenz’s law to determine the “direction” of current and emf.

46.                                                                                                     A square coil of side 5 cm contains 100 loops and is positioned perpendicular to a uniform 0.6 T magnetic field. It is quickly and uniformly pulled from the field (moving  to B) to a region where the field drops abruptly to zero. It takes 0.10 s to remove the coil, whose resistance is 100 . B = 0.6 T 5 cm

47.                                                             Initial: Bi = = BA .                                         (a) Find the change in flux through the coil. Final: Bf = 0 . B = Bf - Bi = 0 - BA = -(0.6 T)(0.05 m)2 = -1.5x10-3 Wb.

48.                                                                                                     (b) Find the current and emf induced. Current will begin to flow when the coil starts to exit the magnetic field. Because of the resistance of the coil, the current will eventually stop flowing after the coil has left the magnetic field.  final initial The current must flow clockwise to induce an “inward” magnetic field (which replaces the “removed” magnetic field).

49.                                           v The induced emf is dA x x

50. “uniformly” pulled The induced current is