Prof. D. Wilton ECE Dept. Notes 11 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston. Gauss.

Prof. D. WiltonECE Dept.

Notes 11

ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism

Notes prepared by the EM group,

University of Houston.

Gauss

ExampleExample

Assume D D

infinite uniform line charge

encl

S

D n dS Q

Find the electric field everywhere

x

y

z

S

l = l0 [C/m]

h

r

Example (cont.)Example (cont.)

0

c

t

b

encl l

S S

S

S

Q h

D n dS D dS

D z dS

D z dS

h

St

Sb

Sc

r

Example (cont.)Example (cont.)

0

0

2

2

2

c cS S S

l

l

D n dS D dS D dS

D h

D h h

D

Hence

So 0

0

V/m2

lE

ExampleExample

v = 3 2 [C/m3] , < a

Assume D D

non-uniform infinite cylinder of volume charge density

encl

S

D n dS Q

x

y

z

S

h

a

r

Find the electric field everywhere

Example (cont.)Example (cont.)(a) < a

2

0 0 0

0

2

0

4

0

4

2

2 3

32

4

3

2

encl v

V

h

v

v

encl

Q dV

d d dz

h d

h d

h

Q h

S

h

r

Example (cont.)Example (cont.)

Hence

So

4

3

2

32

23

4

cS S S

D n dS D n dS D dS

D h

D h h

D

3

0

3V/m

4E a

Example (cont.)Example (cont.)

(b) > a

43

2enclQ h a

432

2D h ha

4

0

3V/m

4

aE a

S

h

r

ExampleExample

x

y

z

l0 -h

-h

When Gauss’s Law is not useful:

!

!

encl

S

encl

D n dS Q

D D

Q h

(3) E has more than one component

But (1)

(2) (the charge density is not uniform!)

ExampleExample

y

z

x

s = s0 [C/m2]

Assume

zD z D z

encl

S

D n dS Q S

A

r

Find the electric field everywhere

2

top

bottom

z encl

S

z

S

z encl

S

z z encl

z z

z encl

D z n dS Q

D z z dS

D z z dS Q

D A D A Q

D D

AD Q

Example (cont.)Example (cont.)

Assume

S

A

r

D

D

z

0

0 0

0

2

2 2

2

encl s

z encl

s sz

sz z

Q A

AD Q

AD

A

D D

Example (cont.)Example (cont.)

so 0

0

[V/m] 0, 02

sE z z z

S

A

r

( Generally, Ez is continuous except on either side of a surface charge)

ExampleExample

slab of uniform charge

0 0

x

x x

x

E x E x

E x E x

E

Assume

(since Ex(x) is a continuous function)

y

x

30 [C/m ]v

d

rFind the electric field everywhere

Example (cont.)Example (cont.)(a) x > d / 2

0

0

0 ( / 2)

/ 2

t b

x encl

S

x x encl

S S

x x encl v

x v

D x n dS Q

D x x dS D x x dS Q

D x A D A Q A d

D x d

0

0

V/m ( / 2)2

v dE x x d

A

S

xxr

30 [C/m ]v

d

Example (cont.)Example (cont.)

Note: If we define

0

0

0

0

V/m2

Note:

so

effs v

effs

effv s

effs v

d

E x

Q Ad A

d

Q

seff

Q

v0

(sheet formula) then

d

A A

Example (cont.)Example (cont.)

(b) 0 < x < d / 2

0

0

x encl v

x v

D A Q A x

D x

y

x

x = 0

x = xS

r

0

0

V/m 0 / 2v xE x x d

30 [C/m ]vd

Example (cont.)Example (cont.)

y

x

d / 2

v0 d / (20 )

x

Ex

- d / 2

0

0

V/m / 2 / 2v xE x d x d

0

0

V/m ( / 2)2

v dE x x d

30 [C/m ]v

Summary

d

0

0

V/m ( / 2)2

v dE x x d