Prof. D. Wilton ECE Dept. Notes 16 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston

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


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


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


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


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


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


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


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


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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Prof. D. Wilton ECE Dept. Notes 16 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston