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

Prof. David R. JacksonECE Dept.

Fall 2014

Notes 17

ECE 2317 Applied Electricity and Magnetism

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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.)

0

0

0

1ˆ lim

1ˆ lim

1ˆ lim

xx

yy

zz

Csx

Csy

Csz

x curl V V drS

y curl V V drS

z curl V V drS

, ,V x y z vector function

curl V vector function

Note: The paths are defined according to the “right-hand rule.”

x

y

z

Cx

Cy

Cz

Sz

Sx

SyCurl 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 of a Vector (cont.)

“Curl meter” ˆ ˆ ˆ, ,x y z

curl V is related to the torque (in the direction).

Assume that V represents the velocity of a fluid.

0

1ˆ limCs

curl V V drS

zshown for

VThe term V dr measures the force on the paddles at each point on the paddle wheel.

C

Torque

Hence

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S

(The relation may be nonlinear, but we are not concerned with this here.)

Curl Calculation

Path Cx :

0, , 02

0, , 02

0, 0,2

0, 0,2

x xx y z zC C

z

y

y

yV dr V dx V dy V dz V z

yV z

zV y

zV y

(1)

(2)

(3)

(4)

Each edge is numbered.

Pair

Pair

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y

z

3

y

z

1 2

4

Cx

The x component of the curl

0, , 0 0, , 02 2

0, 0, 0, 0,2 2

x

x

z z

C

y y

yzx x

yzx

C

y yV V

V dr y zy

z zV V

z yz

VVS S

y z

VVV dr S

y z

Curl Calculation (cont.)

or

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We have multiplied and divided by y.

We have multiplied and divided by z.

x

yzx

C

VVV dr S

y z

0

1ˆ lim

xCsx

x curl V V drS

ˆ yzVV

x curl Vy z

From the curl definition:

Hence

Curl Calculation (cont.)

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From the last slide,

Similarly,

y

z

x zy

C

y xz

C

V VV dr S

z x

V VV dr S

x y

ˆ ˆ ˆy yx xz zV VV VV V

curl V x y zy z z x x y

Hence,

ˆ x zV Vy curl V

z x

ˆ y xV V

z curl Vx y

Curl Calculation (cont.)

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Del Operator

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

x y z

x y z

y yx xz z

V x y z xV yV zVx y z

x y z

x y z

V V V

V VV VV Vx y z

y z x z x y

ˆ ˆ ˆx y zx y z

Recall

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Del Operator (cont.)

curl V V

Hence, in rectangular coordinates,

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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.

ˆ ˆ ˆ ˆˆ ˆ rcurl V r rV V Vr

For example:

Summary of Curl Formulas

1 1ˆˆ ˆz zVV V VV V

V zz z

sin1 1 1 1ˆˆsin sin

r rV rV rVV V V

V rr r r r r

ˆ ˆ ˆy yx xz zV VV VV V

V x y zy z z x x y

Rectangular

Cylindrical

Spherical

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Example

2 2 3ˆ ˆ ˆ3 2 2V x xy z y x z z xz

2 2ˆ ˆ ˆ0 3 3 2 4 6V x z y xy z z x xyz

ˆ ˆ ˆy yx xz zV VV VV V

V x y zy z z x x y

Calculate the curl of the following vector function:

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Example

ˆ ˆ ˆy yx xz zV VV VV V

V x y zy z z x x y

Calculate the curl: ˆV x y

x

y

Velocity of water flowing in a river

ˆ 1V z

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Example (cont.)

ˆ 1V z

ˆ 1 0V z

x

y

Hence

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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.)

ˆS C

V n dS V dr

“The surface integral of circulation per unit area equals the total circulation.”

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C (closed)

S (open)n

ProofDivide S into rectangular patches that are normal to x, y, or z axes

(all with the same area S for simplicity).

ˆ ˆi

irnS

V n dS V n S

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ˆ ˆ ˆ ˆ, ,in x y z orC

S n

S

ˆ inri

Proof (cont.)

ˆ

ˆ ˆ ˆ ˆ, ,

i

i

irC

i

V n S V dr

n x y z

0

1ˆ lim

1

i i

i

i r Cs

C

n V V drS

V drS

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ˆ ˆi

irnS

V n dS V n S

S

C

riˆ in

Ci

Substitute

ˆi

nS C

V n ds V dr so

Proof (cont.)

ˆi

i

nS C

C

C

V n ds V dr

V dr

V dr

exterioredges

Interior edges cancel, leaving only exterior edges.

Proof complete

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S

C

Cancelation

CCi

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ExampleVerify Stokes’s Theorem

ˆV

ˆ

ˆ

1

S

S

S

LHS V n dS

V z dS

VdS

x

= a

y

ABC

C

ˆS C

V n dS V dr

1 1ˆˆ ˆz zVV V VV V

V zz z

S

ˆ ˆ ( )n z RH rule

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Example (cont.)

2

2

2

1

1

12

2

12

4

1

2

S

S

S

S

VLHS dS

dS

dS

dS

a

a

/2

0

/22

0

2

ˆ

ˆ

ˆ

ˆ ˆ

1

2

C

C

A

A

RHS V dr

dr

dr

a dr

a ad

a d

a

Curl Vector Component

S (planar)

l

C

0

1ˆ limS

C

V l V drS

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.

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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)

(proof on next slide) The shape of C is arbitrary. The direction is arbitrary.l

where

Consider the component of the curl vector in an arbitrary direction.

We have:

Also (from continuity):

Hence

ˆS C

V n ds V dr

Stokes’ Theorem:

Proof:

Taking the limit: 0

1ˆ limS

C

V l V drS

ˆC

V l S V dr

ˆˆS

V n ds V l S

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S (planar)

ˆn l (constant)

C

Curl Vector Component (cont.)

Rotation Property of Curl Vector

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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: 0

1ˆ limS

C

V l V drS

Torque

The left-hand side is maximized when the curl vector and the paddle wheel axis are

in the same direction.

ˆ cos cosV l V Torque

V

so

Rotation Property of Curl Vector (cont.)

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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).

V

(axis of whirlpooling)

Rotation Property of Curl (cont.)

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x

y

ˆV x y

ˆ 1V z

Example:

From calculations:

Hence, the paddle wheel spins the fastest when the axis is along the z axis: This is the “whirlpool” axis.

2 22 22 2

0

yx z

A

y yx xz z

AA AV

x y z

V VV VV V

x y x z y x y z z x z y

Vector Identity

Proof:

0V

ˆ ˆ ˆy yx xz zV VV VV V

V x y zy z x z x y

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