ECE 2317
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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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Prof. D. Wilton ECE Dept.

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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 “right-hand 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 “right-hand 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

  • = a,

    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


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