prof. david r. jackson ece dept. fall 2014 notes 17 ece 2317 applied electricity and magnetism 1
TRANSCRIPT
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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.
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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:
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4
x
y
x
y
x
y
=
+
Curl of a Vector (cont.)
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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.)
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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.)
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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
7
y
z
3
y
z
1 2
4
Cx
The x component of the curl
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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
8
We have multiplied and divided by y.
We have multiplied and divided by z.
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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.)
9
From the last slide,
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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:
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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.
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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
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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
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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
19
ˆ ˆi
irnS
V n dS V n S
S
C
riˆ in
Ci
Substitute
ˆi
nS C
V n ds V dr so
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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
20
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
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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:
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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
24
S (planar)
ˆn l (constant)
C
Curl Vector Component (cont.)
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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: 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
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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)
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Rotation Property of Curl (cont.)
27
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.
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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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