Prof. David R. Jackson ECE Dept. Spring 2016 Notes 17 ECE 3318 Applied Electricity and Magnetism 1

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

x y 3 Curl of a Vector (cont.) 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, as seen on the next slide.

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

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

“Curl meter” Assume that V represents the velocity of a fluid. V The term V  dr measures the force on the paddles at each point on the paddle wheel. Torque Hence, 6 SS (If this component is positive, the paddle wheel will spin counterclockwise.)

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

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

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

Similarly, Hence, we have: Curl Calculation (cont.) 10

Del Operator Recall: 11

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

Summary of Curl Formulas Rectangular Cylindrical Spherical 13

Example Calculate the curl of the following vector function: 14

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

Example (cont.) x y Hence 16 Note: The paddle wheel will not spin if the axis is pointed in the x or y directions. The paddle wheel spins opposite to the fingers of the right hand.

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

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

Proof (cont.) 19 S C riri CiCi Substitute so

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

21Example Verify Stokes’s Theorem x  = a y C1C1 C S C2C2 C3C3

22 Example (cont.)

Component of Curl Vector  S (arbitrary planar surface) C 23 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. Consider taking the component of the curl vector in an arbitrary direction. We have the following property:

For the LHS: Hence, Stokes’ Theorem: Proof: Taking the limit: 24  S (planar) C Component of Curl Vector (cont.)

25 The component of the curl vector in any direction tells us what the torque on the paddle wheel will be when we point the axis of the paddle wheel in that direction. Component of Curl Vector (cont.) Physical interpretation of theorem (water flow) We maximize torque by pointing the paddle wheel in the direction of the curl vector. The direction of the curl vector is thus the “axis of the whirlpooling” of the water.

Rotation Property of Curl (cont.) 26 x y Example: From calculations: Hence, the paddle wheel spins the fastest when the axis is along the z axis: This is the axis of the “whirlpool” in the water.

Vector Identity Proof: 27