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
