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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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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:

Hence

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

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:

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

magnetic Gauss law

Ampere’s law

Integral Form of Maxwell’s Equations

electric Gauss law