1. Last Time Magnetic Torque Magnetic Dipole in a B-Field: Potential Energy 1

2. Today Magnetic Force Motors and Generators Gauss' Law 2

3. dF dF  Force on a Magnetic Dipole 3

4. <0 Force on a Magnetic Dipole x Force exerted on dipole by magnetic field 4

5. Two magnets Force on a Magnetic Dipole 5

6. What is the distance at which one magnet can pick up another? =mg  0.1 N Exercise 6

7. B How to Make an Electric Motor Run the current one way B Then reverse the current Torque is in same direction 7

8. Tangential speed: Force on charges: Electric Generators 8

9. AC generator Electric Generators 9

10. Power Required to Turn a Generator 10

11. Demos: 6B-09 Magneto (Electric Generator) 11 3E-12 Steam Engine

12. Building up the Solid Sphere R Total Charge: Total Volume: r What if we stop here at r? Assume: Charge distributed uniformly Charge so far: Volume so far: Charge Volume Charge Volume Uniform charge means these are the SAME 12 Blast from the Past: Lecture 7

13. Where's the Source? Follow the Flow! The source of this fountain is not in the picture. We see the water flow in. We see the same amount flow out. We only see water flowing out. The source of this fountain must be in the bowl. 13

14. What is the Source? The Source of Water is a Water Faucet The Source of E-Field is Charge Water flowing OUT of bowl  there is a Source inside E-field "flowing" out of sphere  there is a Source inside 14

15. Flux through small area: Definition of electric flux on a surface: Electric Flux: Surface Area 15

16. Perpendicular fieldPerpendicular area xx yy Electric Flux: Perpendicular Field or Area q 16

17. Adding up the Flux 17

18. Features: 1. Proportionality constant 2. Size and shape independence 3. Independence on number of charges inside 4. Charges outside contribute zero Gauss’s Law 18

19. What if charge is negative? Works at least for one charge and spherical surface 1. Gauss’s Law: Proportionality Constant 19

20. universe would be much different if exponent was not exactly 2! 2. Gauss’s Law: The Size of the Surface 20

21. Gauss' Law – Size of Surface Point Charge: Integrated over any concentric sphere: ✔ s 21

22. All elements of the outer surface can be projected onto corresponding areas on the inner sphere with the same flux 3. Gauss’s Law: The Shape of the Surface 22

23. – Outside charges contribute 0 to total flux 4. Gauss’s Law: Outside Charges 23

24. 5. Gauss’s Law: Superposition 24

25. iClicker Question One point charge Q is placed inside a closed surface with complicated shape. Which of the following statements is correct on the flux through the surface? a)It’s not possible to calculate the flux since the details of the surface shape are not given. b)The flux is Q/ε 0 c)The location of the charge Q inside the surface is need in order to calculate the flux. d)The flux is not zero and must be negative. +Q

26. iClicker Question Two identical point charges are placed, at the center of a large sphere in one case, and outside an identical sphere in the other case. Which statement about the net electric flux through the surface of the sphere is true? a)The flux is larger when the charge is inside. b)The flux is larger when the charge is outside. c)The flux is the same (and not zero). d)Not enough information to tell. e)The flux is zero in both cases. Q>0

27. Again: Continuous Charge Distribution 1: Charged Line At a point P on perpendicular axis:  x

28. Infinitely long uniformly charged line r h Gauss’s Law: Same result but much less work! E

29. Charged Disk At a point P on axis:

30. Uniformly charged thin, infinite sheet A h Gauss’s Law!

31. Long Cylindrical Capacitor 1.Put charges +q on inner cylinder of radius a, -q on outer cylindrical shell of inner radius b. 2. Calculate E by Gauss’ Law a b +q -q 3. Calculate V 4. Divide q by V C dep. log. on a, b L

32. Spherical Capacitor 1.Put charges +q on inner sphere of radius a, -q on outer shell of inner radius b. 2. Calculate E by Gauss’s Law 3. Calculate V from E 4. Divide q by V q is proportional to V C only depends on a,b (isolated sphere)

33. Today Gauss' Law 33