Prof. D. Wilton ECE Dept. Notes 16 ECE 2317 ECE 2317 Applied Electricity and Applied Electricity and Magnetism Magnetism Notes prepared by the EM group, University of Houston.
Prof. D. Wilton ECE Dept. Notes 16 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston.
Dec 13, 2015
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Prof. D. WiltonECE Dept.
Notes 16
ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism
Notes prepared by the EM group,
University of Houston.
Curl of a VectorCurl of a Vector
0
0
0
1curl lim
1curl lim
1curl lim
x
y
z
CS
CS
CS
x V V drS
y V V drS
z V V drS
, , arbitrary vector functionV x y z
curl vector functionV
x
y
z
Cx
Cy
Cz
S
S
S
Note: Paths are defined according to the “right-hand rule”
, , , ,Cx y z x y z
V drV C
circulationof on
curl,
x Vx
circulation per unit area about etc.
Curl of a Vector (cont.)Curl of a Vector (cont.)
“curl meter” ˆ ˆ ˆ , ,x y z
curl velocityof rotation (in the sense indicated)V
Assume that V represents the velocity of a fluid.
Curl CalculationCurl Calculation
y
z
y
Path Cx :
z 1 2
3
4 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
(side 1)
(side 2)
(side 3)
(side 4)
0, , 0 0, , 02 2
0, 0, 0, 0,2 2
x
x
z z
C
y y
yz
yz
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.)Curl Calculation (cont.)
Though above calculation is for a path about the origin, just add (x,y,z) to all arguments above to obtain the same result for a path about any point (x,y,z) .
x
yz
C
VVV dr S
y z
0
1curl lim
xCsx V V dr
S
curl yzVV
x Vy z
From the curl definition:
Hence
Curl Calculation (cont.)Curl Calculation (cont.)
Similarly,
y
z
x z
C
y x
C
V VV dr S
z x
V VV dr S
x y
curl y yx xz zV VV VV V
V x y zy z z x x y
Hence,
curl x zV Vy V
z x
curl y xV V
z Vx y
Curl Calculation (cont.)Curl Calculation (cont.)
Note the cyclic nature of the three terms:
x
y z
Del OperatorDel Operator
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
x y zx y z
Del Operator (cont.)Del Operator (cont.)
curl V V
Hence,
ExampleExample
2 2 33 2 2V x xy z y x z z xz
2 20 3 2 3 4 6V x z y z xy z x xyz
2 2 33 2 2x y z
x y z x y z
Vx y z x y z
V V V xy z x z xz
ExampleExample
1
y yx xz z
V x y
V VV VV VV x y z
y z x z x y
V z
x
y
Example (cont.)Example (cont.)
1V z
1 0V z
x
y
Summary of Curl FormulasSummary of Curl Formulas
1 1z zVV V VV V
V zz z
sin1 1 1 1
sin sinr r
V rV rVV V VV r
r r r r r
y yx xz zV VV VV V
V x y zy z x z x y
Stokes’s TheoremStokes’s Theorem
n : chosen from “right-hand rule” applied to the surface
S C
V n dS V dr
“The surface integral of circulation per unit area equals the total circulation.”
C
S (open)n
ProofProofDivide S into rectangular patches that are normal to x, y, or z axes.
i ir
iS
V n dS V n S LHS :
Independently consider the left and right hand sides (LHS and RHS) of Stokes’s theorem:
, ,in x y or z
C
S n S
in
ri
iC
Proof (cont.)Proof (cont.)
S
C
, ,
i
i
ir
C
i
V n S V dr
n x y z
e.g ,
0
1lim
ii Cs
n V V drS
i ir
iS
V n dS V n S LHS :
1i
ii r C
n V V drS
Proof (cont.)Proof (cont.)
Hence,
i
i
iri S
i C C
S C
V n S V n ds
V dr V dr
V n ds V dr
(Interior edge integrals cancel)
S
C
C
ExampleExampleVerify Stokes’s theorem for
V x yA B C
x y
C C
C
C C C
V dr V dx V dy
x dy
I I I
0
0A
C
C
C
I
I
( dy = 0 )
x
= a, z= const
y
CA
CB
C
( x = 0 )
CC
(dz = 0)
V x y
Example (cont.)Example (cont.)
2 2 21
0
21
2
sin2 2
sin 12
2 2
a
B
y
y a y a yI
a
a
a
B
B
C
I x dy
2 2
0
a
BI a y dy
x
= a
y
CB
A
B
2
4
aI
Example (cont.)Example (cont.)
Alternative evaluation(use cylindrical coordinates):
2
0
ˆ
B
B
A
B
A
I V dr
V d a d z dz
V a d
cos ,
cos
V V y x x y
x
x a
2
cos cos
cos
V a
a
Now use:
or
Example (cont.)Example (cont.)
Hence
22 2
0
22
0
22
0
2
cos
1 cos2
2
sin 2
2 4
4
BI a d
a d
a
a
2
4
aI
Example (cont.)Example (cont.)Now Use Stokes’s Theorem:
C S
I V dr V z ds
211
4S S
I z z dS dS A a 2
4
aI
V x y y yx xz zV VV VV V
V x y zy z x z x y
1V z
ˆ( )n z
Rotation Property of CurlRotation Property of Curl
(constant)
S (planar)
n
C
0
1limS
C
V n V drS
The component of curl in any direction measures the rotation (circulation) about that direction
Rotation Property of Curl (cont.)Rotation Property of Curl (cont.)
But
Hence
S C
S
C
V n ds V dr
V n ds V n S
V n S V dr
Stokes’s Th.:
Proof:
Taking the limit: 0
1limS
C
V n V drS
(constant)
S (planar)
n
C
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 IdentityVector Identity
0V
y yx xz zV VV VV V
V x y zy z x z x y
Proof:
Vector IdentityVector Identity
Visualization:
0V
1
1ˆ i
iS
nV
V V
ii
Ci
V dr
S
face
0
VV
Flux of out of
iCV ˆ in iS
Edge integrals cancel when summed over closed box!
ExampleExample
Find curl of E:
s0 l0
q
1 2 3
Infinite sheet of charge (side view)
Infinite line charge Point charge
Example (cont.)Example (cont.)
0
0
ˆ2
sE x
y yx xz zE EE EE E
E x y zy z x z x y
0 0 0 0 0 0
0
E x y z
s0
1
x
1 1
0
z zEE E EE E
E zz z
l0
2
0
02E
Example (cont.)Example (cont.)
sin1 1 1 1
sin sin
0
r rE rE rEE E E
E rr r r r r
q
32
04
qE r
r
Example (cont.)Example (cont.)
0E By superposition, the result ,
must be true for any general charge distribution
Faraday’s Law (Differential Form)Faraday’s Law (Differential Form)
0S C
E n dS E dr
Let S S
n
S
0S
E n dS
Hence
Stokes’s Th.:
Let S 0: 0n E S 0n E
small planar surface
(in statics)
Faraday’s Law (cont.)Faraday’s Law (cont.)n
S
0
0
0
x E
y E
z E
0E Hence
0n E
ˆ ˆ ˆLet , , :n x y z
Faraday’s Law (Summary)Faraday’s Law (Summary)
0E
0C
E dr Integral form of Faraday’s law
Differential (point) form of Faraday’s law
Stokes’s theorem
curl definition
Path IndependencePath Independence
0V Assume
A BC1
C2
2
2
C
I V dr 1
1
C
I V dr
1 2I I
Path Independence (cont.)Path Independence (cont.)
Proof
2 1
0C C C S
V d r V n dS
2 1 0I I
A B
C
C = C2 - C1
S is any surface that is attached to C.
(proof complete)
Path Independence (cont.)Path Independence (cont.)
0V
path independence
Stokes’s theorem Definition of curl
0C
V dr
Summary of ElectrostaticsSummary of Electrostatics
0
0vD
E
D E
Faraday’s Law: DynamicsFaraday’s Law: Dynamics
0E In statics,
Experimental Law(dynamics):
BE
t
BE
t
magnetic field Bz (increasing with time)
x
y
electric field E
ˆ 0zBz E
t
(assume that Bz increases with time)
Faraday’s Law: Dynamics (cont.)Faraday’s Law: Dynamics (cont.)
Faraday’s Law: Integral FormFaraday’s Law: Integral Form
BE
t
Apply Stokes’s theorem:
ˆ
ˆ
S C
S
E n dS E dr
Bn dS
t
Faraday’s Law (Summary)Faraday’s Law (Summary)
BE
t
ˆC S
BE dr n dS
t Integral form of Faraday’s law
Differential (point) form of Faraday’s law
Stokes’s Theorem
Faraday’s Law (Experimental Setup)Faraday’s Law (Experimental Setup)
magnetic field B (increasing with time)
x
y+
-V > 0
Note: the voltage drop along the wire is zero
Faraday’s Law (Experimental Setup)Faraday’s Law (Experimental Setup)
x
y+
-V > 0
Note: the voltage drop along the wire is zero
ˆ
0
C S
z
S
BE dr n dS
t
BdS
t
S
C
ˆ( )n z
0V
B
A C
V E dr E dr
Hence
A
B
Differential Form of Differential Form of Maxwell’s EquationsMaxwell’s Equations
0
vD
BE
tB
DH J
t
electric Gauss law
magnetic Gauss law
Faraday’s law
Ampere’s law
Integral Form of Integral Form of Maxwell’s EquationsMaxwell’s Equations
ˆ
ˆ
ˆ 0
ˆ ˆ
v
S V
C S
S
C S S
D n dS dV
dE dr B n dS
dt
B n dS
dH dr J n dS D n dS
dt
electric Gauss law
magnetic Gauss law
Faraday’s law
Ampere’s law
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