Prof. David R. JacksonECE Dept.

Fall 2014

Notes 9

ECE 2317 Applied Electricity and Magnetism

1

Flux Density

q

E

20

ˆ4

qE r

r

Define:

“flux density vector”

2

0D E

120 F/m8.854187818 10

From Coulomb’s law:

We then have2

2ˆ [C/m ]

4

qD r

r

Flux Through Surface

q

D

n̂

S

ˆ CS

D n dS

Define flux through a surface:

3

The electric flux through a surface is analogous to the current flowing through a surface.

ˆ AS

I J n dS The total current (amps) through a surface is the “flux of the current density.”

In this picture, flux is the flux crossing the surface in the upward sense.

Cross-sectional view

Flux Through Surface (cont.)

4

Analogy with electric current

ˆ AS

I J n dS

A small electrode in a conducting medium spews out current equally in all directions.

I

n̂

S

J2

ˆ4

IJ r

r

Cross-sectional view

Flux Through Surface (cont.)

5

22

ˆ [A/m ]4

IJ r

r2

2ˆ [C/m ]

4

qD r

r

J

I

D

q

Analogy

Example

2

2

ˆ

ˆ

ˆ ˆ4

4

S

S

S

S

D n dS

D r dS

qr r dS

r

qdS

r

(We want the flux going out.)

ˆ ˆn r

Find the flux from a point charge going out through a spherical surface.

6

x

y

z

q

D

S

r

Example (cont.)

22

20 0

2

0 0

0

sin4

sin4

2 sin4

2 24

qr d d

r

qd d

qd

q

C[ ]q

7

We will see later that his has to be true from Gauss’s law.

Water Analogy

x

y

z

w

8

Each electric flux line is like a stream of water.

w = flow rate of water [liters/s]

Each stream of water carries w / Nf [liters/s].

Water nozzle

Nf streamlinesWater streams

Note that the total flow rate through the surface is w.

Water Analogy (cont.)

9

Here is a real “flux fountain” (Wortham fountain, on Allen Parkway).

Flux in 2D Problems

l

D

n̂

C

ˆ C/ml

C

D n dl

The flux is now the flux per meter in the z direction.

10

We can also think of the flux through a surface S that is the contour C extruded h in the z direction.

ˆ CS

D n dl h [m]

S

l

C ml h

Flux Plot (2D)

1) Lines are in direction of D

2) #

D flux lines

length

LND

L

L = small length perpendicular to the flux vector

NL = # flux lines through L

l0

DL

x

y

Rules:

11

Rule #2 tells us that a region with a stronger electric field will have flux lines

that are closer together. Flux lines

Example

0

0

V/mˆ2

lE

#D

L

lines

Draw flux plot for a line charge

20 C/mˆ2

lD

0 0

1 1 1

#

2 2lD

C C C

linesconstantHencel0 [C/m]

Nf lines

x

y

L

12

1 1

1

# #

#

D C CL

C

lines lines

lines

Rule 2:

#

lines

constant

l0 [C/m]

13

Example (cont.)

This result implies that the number of flux lines coming

out of the line charge is fixed, and flux lines are thus never

created or destroyed.

This is actually a consequence of Gauss’s law.

(This is discussed later.)

Nf lines

x

y

L

Example (cont.)

Note: If Nf = 16, then each flux line represents l0 / 16 [C/m]

Choose Nf = 16 l0 [C/m]

x

y

Note: Flux lines come out of positive charges and

end on negative charges.

Flux line can also terminate at infinity.

14

#

lines

constant

Flux Property

NC is the number of flux lines through C.C

ˆl C

C

D n dl N

The flux (per meter) l through a contour is proportional to the number of flux lines that cross the contour.

15

Note: In 3D, we would have that the total flux through a surface is proportional to the number of flux lines crossing the surface.

Please see the Appendix for a proof.

Example

Nf = 16

l0 = 1 [C/m]

x

y

C ˆl

C

D n dl

Graphically evaluate

lines /line1

4 C/m16l

C/m1

4l

16

Note: The answer will be more accurate if we use a plot with more flux lines!

Line charge example

l0

D

x

y

17

Equipotential Contours

Equipotential contours CV

= 1 [V]

= -1 [V]

= 0 [V]

Flux lines

The equipotential contour CV is a contour on which the potential is constant.

Equipotential Contours (cont.)

D CV

18

Property:

The flux line are always perpendicular to the equipotential contours.

CV: ( = constant )

CV

D

(proof on next slide)

Equipotential Contours (cont.)

Proof:

0

0

0

E r

D r

D r

On CV :

19

The r vector is tangent (parallel) to the contour CV.

CV

D

A

B

r

B

AB

A

V E dr

E r

Two nearby points on an equipotential contour are considered.

Proof of perpendicular property:

Method of Curvilinear Squares

Assume a constant voltage difference V between adjacent equipotential lines in a

2D flux plot.

2D flux plot

Note: Along a flux line, the voltage always decreases as we go in the direction of the flux line.

0B B B

AB

A A A

V V E dr E dr E dl

20

Note: It is called a curvilinear “square” even though the shape may be rectangular.

(If we integrate along the flux line, E is in the same direction as to dr.)“Curvilinear square”

CV

DV

AB+

-

Method of Curvilinear Squares (cont.)Theorem: The shape (aspect ratio) of the “curvilinear squares” is preserved throughout the plot.

L

Wconstant

CV

DV CV

WL

V

21

Assumption: V is constant throughout plot.

Proof of constant aspect ratio property

B

AB

A

V E dr V

If we integrate along the flux line, E is in the same direction as dr.

so

B

A

E dr VB

A

E dl V E L VTherefore

Method of Curvilinear Squares (cont.)

Hence,

22

CV

D

WL

A

B

V+

-

Also,

VL

E

1 1

1L LN ND C C

L L W

1CW

D

Hence,

01 1 1

D DL V V V

W E C C E C

constant

so

Method of Curvilinear Squares (cont.)

23

Hence,

(proof complete)

CV

D

WL

AB

V+

-

Summary of Flux Plot Rules

24

1) Lines are in direction of D .

2) Equipotential contours are perpendicular to the flux lines.

3) We have a fixed V between equipotential contours.

4) L / W is kept constant throughout the plot.

#D

flux lines

length

If all of these rules are followed, then we have the following:

ExampleLine charge

The aspect ratio L/W has been chosen to be unity in this plot.

This distance between equipotential contours (which defines W) is proportional to the radius (since the distance between flux lines is).

25

l0

D

x

y

L

W

Note how the flux lines get closer as we approach the

line charge: there is a stronger electric field there.

Example

http://www.opencollege.com

26

A parallel-plate capacitor Note: L / W 0.5

Example

Figure 6-12 in the Hayt and Buck book.

Coaxial cable with a square inner conductor

1L

W

27

28

http://en.wikipedia.org/wiki/Electrostatics

Flux lines are closer together where the field is stronger.

The field is strong near a sharp conducting corner.

Flux lines begin on positive charges and end on negative charges.

Flux lines enter a conductor perpendicular to it.

Some observations:

Flux Plot with Conductors (cont.)Conductor

Example of Electric Flux Plot

29

Electroporation-mediated topical delivery of vitamin C for cosmetic applicationsLei Zhanga, , Sheldon Lernerb, William V Rustruma, Günter A Hofmanna

a Genetronics Inc., 11199 Sorrento Valley Rd., San Diego, CA 92121, USAb Research Institute for Plastic, Cosmetic and Reconstructive Surgery Inc., 3399 First Ave., San Diego, CA 92103, USA.

Note: In this example, the aspect ratio L/W is

not held constant.

Example of Magnetic Flux PlotSolenoid near a ferrite core (cross sectional view)

Flux plots are often used to display the

results of a numerical simulation, for either the

electric field or the magnetic field.

30

Magnetic flux lines

Ferrite core Solenoid windings

Appendix: Proof of Flux Property

31

Proof of Flux Property

ˆ cosl D n L D L

NC : flux lines

Through C

cosl D L

l D L

so

or

32

NC : # flux lines

C

D

n̂

L

One small piece of the contour(the length is L)

C

LC

L

D

L

Flux Property Proof (cont.)

Also,

l D L

CND

L

(from the definition of a flux plot)

Hence, substituting into the above equation, we have

Cl C

ND L L N

L

l CN Therefore,

33

L

D

(proof complete)