Prof. David R. Jackson ECE Dept. Fall 2014 Notes 17 ECE 2317 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.

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:

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 located at the point of interest (a separation 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 (The relation may be nonlinear, but we are not concerned with this here.)

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, Curl Calculation (cont.) 10

Del Operator Recall 11

Del Operator (cont.) Hence, in rectangular coordinates, 12 See Appendix A.2 in the Hayt & Buck book for a general derivation 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

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

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 A B C C S

22 Example (cont.)

Curl Vector Component  S (planar) C 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. 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. (proof on next slide)  The shape of C is arbitrary.  The direction is arbitrary. where Consider the component of the curl vector in an arbitrary direction. We have:

Also (from continuity): Hence Stokes’ Theorem: Proof: Taking the limit: 24  S (planar) C Curl Vector Component (cont.)

Rotation Property of Curl Vector 25 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. so

Rotation Property of Curl Vector (cont.) 26 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)

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

Vector Identity Proof: 28