Prof. D. Wilton ECE Dept.

1. ECE 2317 Applied Electricity and Magnetism Prof. D. Wilton ECE Dept. Notes 16 Notes prepared by the EM group, University of Houston.

2. Curl of a Vector z S Cz S y S Cy Note: Paths are defined according to the “right-hand rule” x Cx

3. Curl of a Vector (cont.) “curl meter” Assume that V represents the velocity of a fluid.

4. Curl Calculation z Path Cx : Cx 4 1 z 2 y y 3 (side 1) (side 2) (side 3) (side 4)

5. Curl Calculation (cont.) Though above calculation is for a path about the origin, just add (x,y,z) to all arguments above to obtain the same result for a path about any point (x,y,z) .

6. Curl Calculation (cont.) From the curl definition: Hence

7. Curl Calculation (cont.) Similarly, Hence, x Note the cyclic nature of the three terms: y z

8. Del Operator

9. Del Operator (cont.) Hence,

10. Example

11. Example y x

12. Example (cont.) y x

13. Summary of Curl Formulas

14. S (open) C : chosen from “right-hand rule” applied to the surface Stokes’s Theorem “The surface integral of circulation per unit area equals the total circulation.”

15. S C Proof Divide S into rectangular patches that are normal to x, y, or z axes. Independently consider the left and right hand sides (LHS and RHS) of Stokes’s theorem:

16. S C Proof (cont.)

17. S C C Proof (cont.) Hence, (Interior edge integrals cancel)

18. Example Verify Stokes’s theorem for y • = a, z= const (dz= 0) CB CC C x CA ( dy= 0 ) ( x= 0 )

19. Example (cont.) y B  = a CB x A

20. Example (cont.) Alternative evaluation (use cylindrical coordinates): Now use: or

21. Example (cont.) Hence

22. Example (cont.) Now Use Stokes’s Theorem:

23. (constant) S(planar) C Rotation Property of Curl The component of curl in any direction measures the rotation (circulation) about that direction

24. (constant) S(planar) C Rotation Property of Curl (cont.) Proof: Stokes’s Th.: But Hence Taking the limit:

25. Vector Identity Proof:

26. Vector Identity Visualization: Edge integrals cancel when summed over closed box!

27. Example Find curl of E: 3 2 1 q s0 l0 Infinite sheet of charge (side view) Point charge Infinite line charge

28. Example (cont.) 1 x s0

29. Example (cont.) 2 l0

30. 3 q Example (cont.)

31. Faraday’s Law (Differential Form) (in statics) Stokes’s Th.: small planar surface Hence Let S  0:

32. Faraday’s Law (cont.) Hence

33. Faraday’s Law (Summary) Integral form of Faraday’s law curl definition Stokes’s theorem Differential (point) form of Faraday’s law

34. Path Independence Assume B A C1 C2

35. Path Independence (cont.) Proof B A C C = C2 - C1 S is any surface that is attached to C. (proof complete)

36. Path Independence (cont.) Stokes’s theorem Definition of curl

37. Summary of Electrostatics

38. Faraday’s Law: Dynamics In statics, Experimental Law (dynamics):

39. Faraday’s Law: Dynamics (cont.) (assume thatBz increases with time) y magnetic field Bz (increasing with time) x electric field E

40. Faraday’s Law: Integral Form Apply Stokes’s theorem:

41. Faraday’s Law (Summary) Integral form of Faraday’s law Stokes’s Theorem Differential (point) form of Faraday’s law

42. + V > 0 y - x Note: the voltage drop along the wire is zero Faraday’s Law (Experimental Setup) magnetic field B (increasing with time)

43. Faraday’s Law (Experimental Setup) + A V > 0 y - B C x S Note: the voltage drop along the wire is zero Hence

44. Differential Form of Maxwell’s Equations electric Gauss law Faraday’s law magnetic Gauss law Ampere’s law

45. Integral Form of Maxwell’s Equations electric Gauss law Faraday’s law magnetic Gauss law Ampere’s law