notes16 2317 - University of Houston
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Transcript notes16 2317 - University of Houston
ECE 2317
Applied Electricity and Magnetism
Prof. D. Wilton
ECE Dept.
Notes 16
Notes prepared by the EM group,
University of Houston.
Curl of a Vector
V x, y, z arbitrary vector function
curl V vector function
z
S
Cz
S
y
S
1
S 0 S
1
y curl V lim
S 0 S
1
z curl V lim
S 0 S
x curl V lim
Cx
Cy
Cz
V dr
V dr
V dr
V dr ofcirculation
V on C
Cx, y , z
x, y, z
x curl V circulation per
unit area about
x, etc.
Cy
x
Cx
Note: Paths are defined
according to the “right-hand rule”
Curl of a Vector (cont.)
“curl meter”
xˆ , yˆ , zˆ
Assume that V represents
the velocity of a fluid.
curl V velocityof rotation (in the sense indicated)
Curl Calculation
z
Path Cx :
4
1
z 2
3
Cx
V dr
Cx
y
y
y
V
dx
V
dy
V
dz
V
, 0 z
y
z
z 0,
Cx x
2
y
Vz 0,
, 0 z
2
z
Vy 0, 0, y
2
z
Vy 0, 0,
y
2
(side 1)
(side 2)
(side 3)
(side 4)
Curl Calculation (cont.)
y
y
V
0,
,
0
V
0,
, 0
z
z 2
2
V
dr
y
z
Cx
y
z
z
Vy 0, 0, 2 Vy 0, 0, 2
z y
z
Vy
Vz
S
S
y
z
Vz Vy
C V dr S y z
x
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) .
Curl Calculation (cont.)
Vz Vy
C V dr S y z
x
From the curl definition:
Hence
1
s 0 S
x curl V lim
Vz Vy
x curl V
y
z
Cx
V dr
Curl Calculation (cont.)
Similarly,
Vx Vz
V
dr
S
C
z
x
y
Vx Vz
y curl V
z
x
Vy Vx
C V dr S x y
z
Vy Vx
z curl V
x
y
Hence,
Vz Vy
Vx Vz Vy Vx
curl V x
y
z
z
x x
y
z
y
x
Note the cyclic nature of the three terms:
y
z
Del Operator
x y z
y
z
x
V x y z xVx yVy zVz
y
z
x
x
y
z
x
Vx
y
Vy
z
Vz
Vz Vy
Vz Vx Vy Vx
x
y
z
y
z
x
z
x
y
Del Operator (cont.)
Hence,
curl V V
Example
V x 3xy z y 2 x z z 2 xz
2
V
2
3
x
y
z
x
y
z
x
Vx
y
Vy
z
Vz
x
3xy 2 z
y
2x2 z3
z
2 xz
V x 0 3z
2
y 2 z 3xy z 4 x 6 xyz
2
Example
V x y
Vz Vy
Vz Vx
V x
y
z
z
x
y
Vy Vx
z
y
x
V z 1
y
x
Example (cont.)
V z 1
V z 1 0
y
x
Summary of Curl Formulas
Vz Vy
Vz Vx Vy Vx
V x
y
z
y
z
x
z
x
y
1 Vz V
V Vz
1 V V
V
z
z
z
1 V sin V
V
r sin
1 1 Vr rV
1 rV Vr
r
r sin
r
r r
Stokes’s Theorem
S (open)
n
C
n : chosen from “right-hand rule” applied to the surface
V n dS V dr
S
C
“The surface integral of circulation per unit area equals the total circulation.”
Proof
Divide S into rectangular patches that are normal to x, y, or z axes.
ni
r
i
n
Ci
S
S
n x, y, or z
i
C
Independently consider the left and right
hand sides (LHS and RHS) of Stokes’s theorem:
LHS :
V n dS V
S
i
ri
ni S
Proof (cont.)
S
C
LHS :
V n dS V
1
ni V lim
s 0 S
ni V ri
n i S
i
S
1
S
ri
Ci
Ci
V dr
V dr
V r ni S V dr
i
Ci
e.g, ni x, y, z
Proof (cont.)
S
C
Hence,
V
i
ri
n i S V n ds
S
V dr V dr
i
Ci
C
V n ds V dr
S
C
C
(Interior edge integrals cancel)
Example
y
CC
Verify Stokes’s theorem
for V x y
= a,
z= const
V dr V
CB
C
C
CA
dx V y dy
C
x
x
x dy
C
I C A I CB I CC
V x y
ICA 0
( dy = 0 )
I CC 0
(x=0)
(dz = 0)
Example (cont.)
IB
x dy
a
y a y
a 2 1 y
IB
sin
2
2
a
y 0
2
CB
y
=a
B
CB
x
a 2 1
sin 1
2
a2
2 2
A
a
I B a 2 y 2 dy
0
I
a2
4
2
Example (cont.)
Alternative evaluation
(use cylindrical coordinates):
B
I B V dr
A
B
V d ˆ a d z dz
A
2
V a d
0
Now use:
V V y x x y
x cos ,
x a cos
or
V a cos cos
a cos
2
Example (cont.)
Hence
IB
2
a 2 cos 2 d
0
a2
1 cos2
0 2 d
2
2
sin
2
a2
4 0
2
a
4
2
a 2
I
4
Example (cont.)
Now Use Stokes’s Theorem:
I V dr V z ds
C
V xy
(nˆ z)
S
Vz Vy
Vz Vx Vy Vx
V x
y
z
y
z
x
z
x
y
V z 1
1
I z 1 z dS dS A a 2
4
S
S
I
a2
4
Rotation Property of Curl
n
1
V dr
V n lim
S 0 S
C
(constant)
S (planar)
C
The component of curl in any direction
measures the rotation (circulation) about
that direction
Rotation Property of Curl (cont.)
n
Proof:
Stokes’s Th.:
V n ds V dr
S
But
C
V n ds V n S
S
Hence
(constant)
V n S V dr
C
Taking the limit:
1
V dr
V n lim
S 0 S
C
S (planar)
C
Vector Identity
V 0
Proof:
Vz Vy
Vz Vx Vy Vx
V x
y
z
y
z
x
z
x
y
Ax Ay Az
V
x
y
z
A
2Vz 2Vy 2Vz 2Vx 2Vy 2Vx
x y x z y x y z z x z y
0
Vector Identity
V 0
Visualization:
V
Edge integrals cancel
when summed over
closed box!
nˆ i Si
Ci
Flux of V out of V
1
V
V
face i
V nˆ i Si
1
V dr
Si C
i
0
Example
Find curl of E:
3
2
1
q
s0
Infinite sheet of charge
(side view)
l0
Infinite line charge
Point charge
Example (cont.)
1
x
s0
E xˆ
2
0
s0
Ez E y
Ez Ex E y Ex
E x
y
z
z
z x
y
x
y
E x 0 0 y 0 0 z 0 0
0
Example (cont.)
2
0
E
2
0
l0
1 Ez E
E Ez
1 E E
E
z
z
z
0
Example (cont.)
3
q
1
E
r sin
q
E r
2
4 0 r
E sin E
1 1 Er rE
1 rE Er
r
r
sin
r
r
r
0
By superposition, the result E 0 ,
must be true for any general charge distribution
Faraday’s Law (Differential Form)
Stokes’s Th.:
E n dS E dr 0
S
(in statics)
C
n
Let S S
Hence
S
small planar surface
E n dS 0
S
Let S 0:
n E S 0
n E 0
Faraday’s Law (cont.)
n E 0
Let n xˆ , yˆ , zˆ :
x E 0
y E 0
z E 0
Hence
E 0
n
S
Faraday’s Law (Summary)
E dr 0
Integral form of Faraday’s law
C
Stokes’s
theorem
curl
definition
E 0
Differential (point) form of Faraday’s law
Path Independence
V 0
Assume
A
B
C1
I1 V d r
C2
I2 V d r
C1
C2
I1 I 2
Path Independence (cont.)
Proof
B
A
C
C = C2 - C1
C C2 C1
V d r V n dS 0
S
I 2 I1 0
S is any surface that is
attached to C.
(proof complete)
Path Independence (cont.)
V 0
Stokes’s theorem
Definition of curl
path independence
V dr 0
C
Summary of Electrostatics
D v
E 0
D 0 E
Faraday’s Law: Dynamics
In statics,
E 0
Experimental Law
(dynamics):
B
E
t
Faraday’s Law: Dynamics (cont.)
B
E
t
Bz
zˆ E
0
t
(assume that Bz increases with time)
y
magnetic field Bz (increasing with time)
x
electric field E
Faraday’s Law: Integral Form
B
E
t
Apply Stokes’s theorem:
E nˆ dS E d r
S
C
B
nˆ dS
t
S
Faraday’s Law (Summary)
B
C E d r S t nˆ dS
Integral form of Faraday’s law
Stokes’s Theorem
B
E
t
Differential (point) form of Faraday’s law
Faraday’s Law (Experimental Setup)
+
y
V >0
-
x
Note: the voltage drop
along the wire is zero
magnetic field B (increasing with time)
Faraday’s Law (Experimental Setup)
+
A
y
B
C E d r S t nˆ dS
Bz
dS 0
t
S
V >0
-
B
C
x
S
B
V E dr
A
Hence
E dr
C
V 0
(nˆ z)
Note: the
voltage drop
along the wire
is zero
Differential Form of
Maxwell’s Equations
D v
B
E
t
B 0
D
H J
t
electric Gauss law
Faraday’s law
magnetic Gauss law
Ampere’s law
Integral Form of
Maxwell’s Equations
D nˆ dS dV
v
S
electric Gauss law
V
d
C E d r dt S B nˆ dS
B nˆ dS 0
Faraday’s law
magnetic Gauss law
S
d
C H d r S J nˆ dS dt S D nˆ dS
Ampere’s law