Prof. David R. Jackson ECE Dept.

1. ECE 2317 Applied Electricity and Magnetism Fall 2014 Prof. David R. Jackson ECE Dept. Notes 17

2. Curl of a Vector The curl of a vector function measures the tendency of the vector function to circulate or rotate (or “curl”) about an axis. Note the circulation about the z axis in this stream of water. y x

3. Curl of a Vector (cont.) y x Here the water also has a circulation about the z axis. This is more obvious if we subtract a constant velocity vector from the water:

4. Curl of a Vector (cont.) y x = y x + y x

5. Curl of a Vector (cont.) z Sz Cz Sy y Sx Cy x Cx It turns out that the results are independent of the shape of the paths, but rectangular paths are chosen for simplicity. Curl is calculated here Note: The paths are defined according to the “right-hand rule.” The paths are all located at the point of interest (a separation is shown for clarity).

6. Curl of a Vector (cont.) “Curl meter” Assume that V represents the velocity of a fluid. Torque S V The termV dr measures the force on the paddles at each point on the paddle wheel. Hence (The relation may be nonlinear, but we are not concerned with this here.)

7. Curl Calculation z The x component of the curl Each edge is numbered. Path Cx : 4 Cx 2 1 y z 3 y (1) (2) (3) (4) Pair Pair

8. Curl Calculation (cont.) We have multiplied and divided by y. We have multiplied and divided by z. or

9. Curl Calculation (cont.) From the last slide, From the curl definition: Hence

10. Curl Calculation (cont.) Similarly, Hence,

11. Del Operator Recall

12. Del Operator (cont.) Hence, in rectangular coordinates, Note: The del operator is only defined in rectangular coordinates. For example: See Appendix A.2 in the Hayt & Buck book for a general derivation that holds in any coordinate system.

13. Summary of Curl Formulas Rectangular Cylindrical Spherical

14. Example Calculate the curl of the following vector function:

15. Example y x Velocity of water flowing in a river Calculate the curl:

16. Example (cont.) y x Note: The paddle wheel will not spin if the axis is pointed in the x or y directions. Hence

17. Stokes’s Theorem S (open) C (closed) The unit normal is chosen from a “right-hand rule” according to the direction along C. “The surface integral of circulation per unit area equals the total circulation.”

18. Proof Divide S into rectangular patches that are normal to x, y, or zaxes (all with the same area S for simplicity). S ri C

19. Proof (cont.) S C Ci ri

20. Proof (cont.) S C Hence, Cancelation C Ci Interior edges cancel, leaving only exterior edges. Proof complete

21. Example Verify Stokes’s Theorem y  = a C B A S x C

22. Example (cont.)

23. Rotation Property of Curl Torque Property of Curl Vector Component S(planar) where (proof on next slide) • The shape of C is arbitrary. • The direction is arbitrary. C (proof on next slide) If we point the "curl meter" in any direction, the torque (in the right-hand sense) (i.e., how fast the paddle rotates) corresponds to the component of the curl in that direction. Note: This property is obviously true for the x, y, and z directions, due to the definition of the curl vector. This theorem now says that the property is true for any direction in space.

24. Rotation Property of Curl (cont.) (constant) Proof: S(planar) Stokes’ Theorem: C Also (from continuity): Hence Taking the limit:

25. Rotation Property of Curl (cont.) Maximizing the Torque on the Paddle Wheel We maximize the torque on the paddle wheel (i.e. how fast it spins) when we point the axis of the paddle in the direction of the curl vector. Proof: The left-hand side is maximized when the curl vector and the paddle wheel axis are in the same direction.

26. Rotation Property of Curl (cont.) To summarize: 1) The component of the curl vector in any direction tells us how fast the paddle wheel will spin if we point it in that direction. 2) The curl vector tells us the direction to point the paddle wheel in to make it spin as fast as possible (the axis of rotation of the “whirlpooling” in the vector field). (axis of whirlpooling)

27. Rotation Property of Curl (cont.) y x Example: From calculations: Hence, the paddle wheel spins the fastest when the axis is along the z axis: This is the “whirlpool” axis.

28. Vector Identity Proof:

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

30. Example (cont.) 1 x s0

31. Example (cont.) 2 l0

32. Example (cont.) 3 q

33. Example (cont.) Note: If the curl of the electric field is zero for the field from a point charge, then by superposition it must be zero for the field from any charge density. This gives us Faraday’s law: (in statics)

34. Faraday’s Law in Statics (Integral Form) C Stokes's theorem: Here Sis any surface that is attached to C. Hence

35. Faraday’s Law in Statics (Differential Form) We show here how the integral form also implies the differential form. Assume We then have: Hence

36. Faraday’s Law in Statics (Summary) Differential (point) form of Faraday’s law Definition of curl Stokes’s theorem Integral form of Faraday’s law

37. Path Independence and Faraday’s Law C2 B A C C1 The integral form of Faraday’s law is equivalent to path independenceof the voltage drop calculation. Proof Hence

38. Summary of Path Independence Equivalent Equivalent properties of an electrostatic field

39. Summary of Electrostatics Here is a summary of the important equations related to the electric field in statics.

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

41. Faraday’s Law: Dynamics (cont.) Assume a Bz that increases with time. y The changing magnetic field produces an electric field. Experiment Magnetic field Bz (increasing with time) x Electric field E

42. Faraday’s Law: Integral Form Integrate both sides over an arbitrary open surface (bowl) S: Apply Stokes’ theorem for the LHS: Note: The right-hand rule determines the direction of the unit normal, from the direction along C. Faraday's law in integral form

43. Faraday’s Law (Experimental Setup) + y v(t)> 0 - x We measure a voltage across a loop due to a changing magnetic field inside the loop. Open-circuited loop Note: The voltage drop along the wire is zero. Magnetic field B(Bz is increasing with time)

44. Faraday’s Law (Experimental Setup) + A y v(t)> 0 - B Also C x S Note: The voltage drop along the wire is zero. So we have

45. Faraday’s Law (Experimental Setup) + A y v(t)> 0 - B C x A = area of loop Assume a uniform magnetic field for simplicity (at least uniform over the loop area). Assume so

46. Lenz’s Law + A y v(t)> 0 - B C x A = area of loop This is a simple rule to tell us the polarity of the output voltage. The voltage polarity is such that it would correspond to a current flow that would oppose the change in flux in the loop (if there were a resistor connected to the output so that current is allowed to flow) . Assume Note: A right-hand rule tells us the direction of the magnetic field due to a wire carrying a current. (A wire carrying a current in the z direction produces a magnetic field in the positive  direction.) Bz is increasing with time.

47. Example: Magnetic Field Probe A small loop can be used to measure the magnetic field (for AC). + y v(t) - Assume C Assume x A = area of loop Then we have At a given frequency, the output voltage is proportional to the strength of the magnetic field.

48. Applications of Faraday’s Law Faraday’s law explains: • How electric generators work • How transformers work Output voltage of generator Output voltage on secondary of transformer Note: For N turns we have

49. The world's first electric generator! (invented by Michael Faraday) A magnet is slid in and out of the coil, resulting in a voltage output. (Faraday Museum, London)

50. The world's first transformer! (invented by Michael Faraday) The primary and secondary coils are wound together on an iron core. (Faraday Museum, London)