Class 31

1. Class 31 • Today we will: • learn about EMF • learn how Faraday’s law works • learn Lenz’s Law and how to apply it

2. Last Time -- Induced Current Accelerating charges produce electric fields in the opposite direction to the acceleration. i i

3. Faraday’s Law If the number of magnetic field lines through a loop is changing, we produce a looping electric field.

4. Induced Current Current increases in a wire… i

5. Induced Current … so the magnetic field increases… i

6. Induced Current … so the number of magnetic field lines passing through the loop (flux) increases… i

7. Induced Current … so there is an induced EMF around the loop … EMF i

8. Induced Current … so current flows around the loop. EMF i i

9. Faraday’s Law of Induction

10. Faraday’s Law of Induction =EMF

11. What is EMF? 1) Any voltage, as from a battery. 2) An effective voltage produced by induced electric fields.

12. What is EMF? 1) Any voltage, as from a battery. 2) An effective voltage produced by induced electric fields. I usually reserve the term EMF for induced voltages.

13. Two Ways to Produce Induced EMF • Acceleration of charges – changing current in a circuit. • Motional EMF – charges in a conductor moving in a B field.

14. Motional EMF A wire of length L moves through a magnetic field. The wire is perpendicular to B. What happens?

15. Motional EMF Charges along the wire feel a Lorentz Force.

16. Motional EMF Charges doesn’t increase indefinitely. Eventually a voltage develops across the wire.

17. Motional EMF Add three other fixed wires to make a loop. Now current will continue to flow around the loop.

18. Clicker Question 1 What happens if all sides of the loop move together? A. Current flows. B. Current doesn’t flow.

19. i i What happens both ways?

20. i i What happens both ways? The magnetic flux – the number of magnetic field lines – passing through the loop changes.

21. Faraday’s Law of Induction Faraday’s Law of Induction …works for both motional EMF and the EMF of accelerating charges!

22. What is EMF? We can think of an induced EMF as a voltage produced all along a wire segment.

23. What is EMF? Let’s take a square loop with an increasing magnetic field passing through it. Assume the wire has a small resistance. V

24. What is EMF? Assume that the EMF around the loop is 40 V. What would a voltmeter read? V

25. The voltmeter would read zero! What is EMF? V = 0V

26. The voltmeter would read zero because the voltage drop due to resistance in the wire segment is exactly the same as the voltage increase due to induction in the wire segment. What is EMF? V = 0V

27. Another way of putting it is that the wire segment is like a lot of little batteries and resistors in series. The voltage goes up through each battery, but down by the same amount through each resistor. What is EMF? V = 0V

28. Each electron that goes around the full loop once gains 40eV of energy from the EMF and loses 40eV of energy to heat! What is EMF? V =0V

29. What if there’s a resistor in the loop? The total EMF is 40V.

30. What if there’s a resistor in the loop? The total EMF is 40V. Ohm’s Law gives i = 8A. The voltage across the resistor is 40V.

31. Where is the EMF being produced?

32. Where is the EMF being produced? Everywhere, including through the resistor.

33. Where is the voltage dropping?

34. Where is the voltage dropping? Primarily in the resistor – just a little in the wire. The resistor only lets a little current trickle through the wire – as compared to having no resistor.

35. Lenz’s Law To determine the direction induced current will flow in a circuit or to determine the direction of the induced electric field, we use Lenz’s Law.

36. Lenz’s Law First, we ask two questions: 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing?

37. Lenz’s Law 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing? 3) Find the direction of the induced magnetic field. The induced magnetic field opposes change in the external magnetic field.

38. Lenz’s Law 1) What is the direction of the external B field? 2) Is the external B field increasing or decreasing? 3) Find the direction of the induced magnetic field. The induced magnetic field opposes change in the external magnetic field. 4) Find the direction the induced currentusing the right-hand rule.

39. Lenz’s Law • The external B is into the screen and increasing. x x x x x x x x x x x x

40. Lenz’s Law • The external B is into the screen and increasing. • To oppose change, the induced B, must be out of the screen. x x x x x x x x x x x x

41. Lenz’s Law • The external B is into the screen and increasing. • To oppose change, the induced B, must be out of the screen. • To produce this B, the current is ccw. x x x x x x x x x x x x i

42. Lenz’s Law • The external B is into the screen and decreasing. x x x x x x x x x x x x

43. Lenz’s Law • The external B is into the screen and decreasing. • To oppose change, the induced B, must be into • the screen. x x x x x x x x x x x x i

44. Lenz’s Law • The external B is into the screen and decreasing. • To oppose change, the induced B, must be into • the screen. • To produce this B, the current is cw. x x x x x x x x x x x x i

45. Class 32 • Today we will: • work several Faraday’s law problems • learn about Eddy currents

46. Maxwell’s Equations in Integral Form • Gauss’s Law of Electricity • Gauss’s Law of Magnetism • Ampere’s Law • Faraday’s Law

47. Maxwell’s Equations in Integral Form • Gauss’s Law of Electricity • Gauss’s Law of Magnetism • Ampere’s Law • Faraday’s Law Flux through a Gaussian surface

48. Maxwell’s Equations in Integral Form • Gauss’s Law of Electricity • Gauss’s Law of Magnetism • Ampere’s Law • Faraday’s Law Flux through a Gaussian surface Line integral around an Amperian loop

49. Maxwell’s Equations in Integral Form • Gauss’s Law of Electricity • Gauss’s Law of Magnetism • Ampere’s Law • Faraday’s Law Flux through a Gaussian surface Line integral around an Amperian loop Flux through the Amperian loop

50. Calculating Flux through a Loop We will assume that the magnetic field is constant over a loop. Then, the flux is: Remember that points in the direction of the normal to the loop.