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# Prof. D. Wilton ECE Dept. - PowerPoint PPT Presentation

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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Applied Electricity and Magnetism

Prof. D. Wilton

ECE Dept.

Notes 16

Notes prepared by the EM group, University of Houston.

z

S

Cz

S

y

S

Cy

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

x

Cx

“curl meter”

Assume that V represents the velocity of a fluid.

z

Path Cx :

Cx

4

1

z

2

y

y

3

(side 1)

(side 2)

(side 3)

(side 4)

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) .

From the curl definition:

Hence

Similarly,

Hence,

x

Note the cyclic nature of the three terms:

y

z

Hence,

y

x

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.”

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:

C

Proof (cont.)

C

C

Proof (cont.)

Hence,

(Interior edge integrals cancel)

Verify Stokes’s theorem

for

y

• = a,

z= const

(dz= 0)

CB

CC

C

x

CA

( dy= 0 )

( x= 0 )

y

B

 = a

CB

x

A

Alternative evaluation

(use cylindrical coordinates):

Now use:

or

Hence

Now Use Stokes’s Theorem:

S(planar)

C

Rotation Property of Curl

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

S(planar)

C

Rotation Property of Curl (cont.)

Proof:

Stokes’s Th.:

But

Hence

Taking the limit:

Proof:

Visualization:

Edge integrals cancel when summed over closed box!

Find curl of E:

3

2

1

q

s0

l0

Infinite sheet of charge

(side view)

Point charge

Infinite line charge

1

x

s0

2

l0

q

Example (cont.)

(in statics)

Stokes’s Th.:

small planar surface

Hence

Let S  0:

curl

definition

Stokes’s

theorem

Differential (point) form of Faraday’s law

Assume

B

A

C1

C2

Proof

B

A

C

C = C2 - C1

S is any surface that is attached to C.

(proof complete)

Stokes’s theorem

Definition of curl

In statics,

Experimental Law

(dynamics):

(assume thatBz increases with time)

y

magnetic field Bz (increasing with time)

x

electric field E

Apply Stokes’s theorem:

Stokes’s Theorem

Differential (point) form of Faraday’s law

V > 0

y

-

x

Note: the voltage drop along the wire is zero

magnetic field B (increasing with time)

+

A

V > 0

y

-

B

C

x

S

Note: the voltage drop along the wire is zero

Hence

electric Gauss law