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. z. S. C z. S. y. S. C y. Note: Paths are defined according to the “righthand rule”. x. C x. Curl of a Vector (cont.).
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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
z
S
Cz
S
y
S
Cy
Note: Paths are defined according to the “righthand rule”
x
Cx
Curl of a Vector (cont.)
“curl meter”
Assume that V represents the velocity of a fluid.
Curl Calculation
z
Path Cx :
Cx
4
1
z
2
y
y
3
(side 1)
(side 2)
(side 3)
(side 4)
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) .
Curl Calculation (cont.)
From the curl definition:
Hence
Curl Calculation (cont.)
Similarly,
Hence,
x
Note the cyclic nature of the three terms:
y
z
Del Operator
Del Operator (cont.)
Hence,
Example
Example
y
x
Example (cont.)
y
x
Summary of Curl Formulas
S (open)
C
: chosen from “righthand rule” applied to the surface
Stokes’s Theorem
“The surface integral of circulation per unit area equals the total circulation.”
S
C
Proof
Divide S into rectangular patches that are normal to x, y, or z axes.
Independently consider the left and right
hand sides (LHS and RHS) of Stokes’s theorem:
S
C
Proof (cont.)
S
C
C
Proof (cont.)
Hence,
(Interior edge integrals cancel)
Example
Verify Stokes’s theorem
for
y
z= const
(dz= 0)
CB
CC
C
x
CA
( dy= 0 )
( x= 0 )
Example (cont.)
y
B
= a
CB
x
A
Example (cont.)
Alternative evaluation
(use cylindrical coordinates):
Now use:
or
Example (cont.)
Hence
Example (cont.)
Now Use Stokes’s Theorem:
(constant)
S(planar)
C
Rotation Property of Curl
The component of curl in any direction measures the rotation (circulation) about that direction
(constant)
S(planar)
C
Rotation Property of Curl (cont.)
Proof:
Stokes’s Th.:
But
Hence
Taking the limit:
Vector Identity
Proof:
Vector Identity
Visualization:
Edge integrals cancel when summed over closed box!
Example
Find curl of E:
3
2
1
q
s0
l0
Infinite sheet of charge
(side view)
Point charge
Infinite line charge
Example (cont.)
1
x
s0
Example (cont.)
2
l0
3
q
Example (cont.)
Faraday’s Law (Differential Form)
(in statics)
Stokes’s Th.:
small planar surface
Hence
Let S 0:
Faraday’s Law (cont.)
Hence
Faraday’s Law (Summary)
Integral form of Faraday’s law
curl
definition
Stokes’s
theorem
Differential (point) form of Faraday’s law
Path Independence
Assume
B
A
C1
C2
Path Independence (cont.)
Proof
B
A
C
C = C2  C1
S is any surface that is attached to C.
(proof complete)
Path Independence (cont.)
Stokes’s theorem
Definition of curl
Summary of Electrostatics
Faraday’s Law: Dynamics
In statics,
Experimental Law
(dynamics):
Faraday’s Law: Dynamics (cont.)
(assume thatBz increases with time)
y
magnetic field Bz (increasing with time)
x
electric field E
Faraday’s Law: Integral Form
Apply Stokes’s theorem:
Faraday’s Law (Summary)
Integral form of Faraday’s law
Stokes’s Theorem
Differential (point) form of Faraday’s law
+
V > 0
y

x
Note: the voltage drop along the wire is zero
Faraday’s Law (Experimental Setup)
magnetic field B (increasing with time)
Faraday’s Law (Experimental Setup)
+
A
V > 0
y

B
C
x
S
Note: the voltage drop along the wire is zero
Hence
Differential Form of Maxwell’s Equations
electric Gauss law
Faraday’s law
magnetic Gauss law
Ampere’s law
Integral Form of Maxwell’s Equations
electric Gauss law
Faraday’s law
magnetic Gauss law
Ampere’s law