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

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

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

4. Curl Calculation y z yy Path C x : zz 1 2 3 4 CxCx (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. From the curl definition: Hence Curl Calculation (cont.)

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

8. Del Operator

9. Del Operator (cont.) Hence,

10. Example

11. Example x y

12. Example (cont.) x y

13. Summary of Curl Formulas

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

15. 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: C S

16. Proof (cont.) S C

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

18. Example Verify Stokes’s theorem for ( dy = 0 ) x  = a, z= const y CACA CBCB C ( x = 0 ) C ( dz = 0 )

19. Example (cont.) x  = a y CBCB A B

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

21. Example (cont.) Hence

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

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

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

25. Vector Identity Proof:

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

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

28. Example (cont.) s0s0 1 x

29. l0l0 2

30. q 3

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

32. Faraday’s Law (cont.) Hence

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

34. Path Independence Assume A B C1C1 C2C2

35. Path Independence (cont.) Proof A B C C = C 2 - C 1 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. magnetic field B z (increasing with time) x y electric field E ( assume that B z increases with time) Faraday’s Law: Dynamics (cont.)

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

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

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

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

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

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