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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. C z.  S. y.  S. C y. Note: Paths are defined according to the “right-hand rule”. x. C x. Curl of a Vector (cont.).

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slide1
ECE 2317

Applied Electricity and Magnetism

Prof. D. Wilton

ECE Dept.

Notes 16

Notes prepared by the EM group, University of Houston.

slide2
Curl of a Vector

z

S

Cz

S

y

S

Cy

Note: Paths are defined according to the “right-hand rule”

x

Cx

slide3
Curl of a Vector (cont.)

“curl meter”

Assume that V represents the velocity of a fluid.

slide4
Curl Calculation

z

Path Cx :

Cx

4

1

z

2

y

y

3

(side 1)

(side 2)

(side 3)

(side 4)

slide5
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) .

slide6
Curl Calculation (cont.)

From the curl definition:

Hence

slide7
Curl Calculation (cont.)

Similarly,

Hence,

x

Note the cyclic nature of the three terms:

y

z

slide14
S (open)

C

: chosen from “right-hand rule” applied to the surface

Stokes’s Theorem

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

slide15
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:

slide16
S

C

Proof (cont.)

slide17
S

C

C

Proof (cont.)

Hence,

(Interior edge integrals cancel)

slide18
Example

Verify Stokes’s theorem

for

y

  • = a,

z= const

(dz= 0)

CB

CC

C

x

CA

( dy= 0 )

( x= 0 )

slide19
Example (cont.)

y

B

 = a

CB

x

A

slide20
Example (cont.)

Alternative evaluation

(use cylindrical coordinates):

Now use:

or

slide22
Example (cont.)

Now Use Stokes’s Theorem:

slide23
(constant)

S(planar)

C

Rotation Property of Curl

The component of curl in any direction measures the rotation (circulation) about that direction

slide24
(constant)

S(planar)

C

Rotation Property of Curl (cont.)

Proof:

Stokes’s Th.:

But

Hence

Taking the limit:

slide26
Vector Identity

Visualization:

Edge integrals cancel when summed over closed box!

slide27
Example

Find curl of E:

3

2

1

q

s0

l0

Infinite sheet of charge

(side view)

Point charge

Infinite line charge

slide30
3

q

Example (cont.)

slide31
Faraday’s Law (Differential Form)

(in statics)

Stokes’s Th.:

small planar surface

Hence

Let S  0:

slide33
Faraday’s Law (Summary)

Integral form of Faraday’s law

curl

definition

Stokes’s

theorem

Differential (point) form of Faraday’s law

slide34
Path Independence

Assume

B

A

C1

C2

slide35
Path Independence (cont.)

Proof

B

A

C

C = C2 - C1

S is any surface that is attached to C.

(proof complete)

slide36
Path Independence (cont.)

Stokes’s theorem

Definition of curl

slide38
Faraday’s Law: Dynamics

In statics,

Experimental Law

(dynamics):

slide39
Faraday’s Law: Dynamics (cont.)

(assume thatBz increases with time)

y

magnetic field Bz (increasing with time)

x

electric field E

slide40
Faraday’s Law: Integral Form

Apply Stokes’s theorem:

slide41
Faraday’s Law (Summary)

Integral form of Faraday’s law

Stokes’s Theorem

Differential (point) form of Faraday’s law

slide42
+

V > 0

y

-

x

Note: the voltage drop along the wire is zero

Faraday’s Law (Experimental Setup)

magnetic field B (increasing with time)

slide43
Faraday’s Law (Experimental Setup)

+

A

V > 0

y

-

B

C

x

S

Note: the voltage drop along the wire is zero

Hence

slide44
Differential Form of Maxwell’s Equations

electric Gauss law

Faraday’s law

magnetic Gauss law

Ampere’s law

slide45
Integral Form of Maxwell’s Equations

electric Gauss law

Faraday’s law

magnetic Gauss law

Ampere’s law

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