Prof. D. WiltonECE Dept.

Notes 9

ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism

Notes prepared by the EM group,

University of Houston.

Electric Flux DensityElectric Flux Density

20

0

22

4

[C/m ]4

qE r

r

D E

qD r

r

Define:

“flux density vector”

q

E

Analogy with Current Flux DensityAnalogy with Current Flux Density

I

J

22

[A/m ]4

IJ r

r

current flux density vector dueto a point source of current

r

r̂ 22

[A/m ]4r

IJ

r

The same current I passes through everysphere concentric with the source, hence

Note: if I is negative, flux density vector points towards I

Current Flux Through SurfaceCurrent Flux Through SurfaceJ

n

S

S

I J n dS A I

Electric Flux Through SurfaceElectric Flux Through Surface

q

D

n

S

S

D n dS C

ExampleExample

2

2

4

4

S

S

S

S

D n dS

D r dS

qr r dS

r

qdS

r

(We want the flux going out)

n r

x

y

z

q

D

S

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

Spherical Surface (cont.)Spherical Surface (cont.)

22

20 0

2

0 0

0

sin4

sin4

2 sin4

2 24

qr d d

r

qd d

qd

q

[C]q

3D Flux Plot for a Point 3D Flux Plot for a Point ChargeCharge

Flux Plot (3D)Flux Plot (3D)Rules:

1) Flux lines are in direction of D

2) #flux lines

areaD

SND

S

NS = # flux lines through S

S

D

S = small area perpendicular to the flux vector

Flux Plot (2D)Flux Plot (2D)Rules:

1) Flux lines are in direction of D

2) #flux lines in plane #flux linesalong

length

#flux lines(in plane)(since # flux linesalong )

length

xy zD

z

xyz z

LND

L

l0

DL

L = small length perpendicular to the flux vector

NL = # flux lines through L

Note: We can construct a 3D problem by

extending the contour in the z direction by one meter to create a surface.

ExampleExample

0

0

V/m2

lE

1

1

0

#lines #lines

#lines

2

D CL L

C

Draw flux plot for a line charge

20 C/m2

lD

0

1

#lines= constant

2 C

Hence

Nf lines

l0 [C/m]

x

y

L

Example (cont.)Example (cont.)

Choose Nf = 16

l0 [C/m]

x

y

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

Flux PropertyFlux Property

NS : flux lines

Through SS

S

S

D n dS N

• The flux through a surface is proportional to the number of flux lines in the flux plot that cross the surface (3D) or contour (2D).

• Flux lines begin on positive charges (or infinity) and end on negative charges (or infinity)

Flux Property (Proof)Flux Property (Proof)

cosD n S D S

cosD S

D S

NS : # flux lines

S

D

n

NS : flux lines

Through SS

DS

S

DS

Flux Property Proof (cont.)Flux Property Proof (cont.)

Also,

S

D D S

SND

S

(from the definition of a flux plot)

Hence SN

SN Therefore,

ExampleExample

Nf = 16

l0 = 1 [C/m]

z = 1 [m] for surface S

x

y

S S

D n dS Find

14 lines C/lines

16

1

4C

Equipotential Surfaces (Contours)Equipotential Surfaces (Contours)

D CV

Proof:

0

0

0

0

PE

F dr

E dr

D dr

On CV :

CV: (V = constant )

dr

CV

D

Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)

CV

D Assume a constant voltage difference V between adjacent equipotential lines in a 2D flux plot.

Theorem: shape of the “curvilinear squares” is preserved throughout the plot.

“curvilinear square”

2D flux plot

Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)

Proof:

CV

D

WL

A

B

B

A

E dr V

Along flux line, E is parallel to dr

Hence,

B

A

E dr VB

A

E dr V E L VOr

Equipotential Surfaces (cont.)Equipotential Surfaces (cont.)

Also,

VL

E

1 1

1L LN ND C C

L L W

1CW

D

Hence,0

1 1

constantDL V V

W C E C

so

CV

D

WL

AB

E L V

ExampleExample

Line charge

l0

D

x

y

ExampleExampleFlux plot for two line charges

hx

y

h

R1 R2

r = (x, y)

l0-l0

0

0 2 22 22l

x x h y y x x h y yD E

x h y x h y

flux lines

- - - - - - - - - - - equipotential lines

line charges of opposite sign line charges of opposite sign

line charges of same sign line charges of same sign

Example Example

0 1 [C/m]l

32fN

Find the flux through the red surface indicated on the figure (z = 1 m)

+ -

Counting flux lines:

Example Example

0 1 [C/m]l

32fN

+ -

1/ 32 [ / flux line]

2 flux lines

C

1/16 [ ]C

Example Example

Software for calculating cross-sectional view of 3D flux plot for two point charges: http://www.xmission.com/~locutus/astro2-old/ElectricField/ElectricField.html