Due to Continuous Charge Distributions
Due to Continuous Charge Distributions Continuous Charge Distributions
Due to a Charged Rod
Due to a Ring of Charge
Due to a Disk of Charge
Electric Field Lines
Homework
Due to Continuous Charge Distribution The electric field at
due to the element of charge
is
The total electric field at due to all the elements in the charge distribution is approximately
Due to Continuous Charge Distribution (cont’d) The total electric field due to the continuous charge distribution is +
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(
)
(*
Charge Densities Linear charge density, , is defined as charge per unit length (units are coulombs per meter)
,
–
If a total charge length , then .
,
is uniformly distributed along a line of -
-
/0.
Surface charge density, , is defined as charge per unit area (units are coulombs per square meter)
1
–
If a total charge area , then
2
1
is uniformly distributed on a surface of -
-
/32
Volume charge density, , is defined as charge per unit volume (units are coulombs per cubic meter) 4
–
If a total charge volume , then 5
4
is uniformly distributed throughout a -
-
/65
Due to a Charged Rod
8
7=
7
>@?
>@?
8
7 :
9
> A
.CBED *
>@? D
8
7 :
9
8 >
A
KML L .CBFD L :
9
A
.CBFD
GH
D
J H
* HI
>@?
N >
D
8
9
VXW :
8 W
SUT :
9
R
VXW R
W
W A
OP J P
J SUT
PQ R
R 9
W A PQ
9 A
S
Z
R
If R
_
Z
:
S[T R]\
S
S[T R
8
S[T R
P
Y
:
8
OP
then
`
Z
R^\
9;: -
Dba
From far away the rod looks like a point charge!
R]\
Y
Due to a Ring of Charge
Figure 19.15
8
7=< :
9
8
7]d A
: 9
7
c"?
e
8
7
fhgjiUk
c"?
8
7]d A
: 9
fhgji[k
7 l T >@?
fhgji[k
8
8 >
>
T c
>@?
e
R=?
: 9
R=?nm
8
7]d A
>
7
l T >
R ?
mpo ?
? /
9;:
8 e
e
8
A >
7
7]d
*
* l T >
9 e
R ?
?
:
8
l
A
8
ZrqtsuR^\
T >@?
: 9
>
R=?vm
o
/
?
? /
mpo
>
l T >@?
R=?nm
o
/
?
Due to a Ring of Charge
Figure 19.15 : 9
e
8 >
l T >
R ?
?nm
`
o
/
?
9;: e
If >
_
R
then -
a
From far away the ring looks like a point charge!
Due to a Disk of Charge dr r
dE
P
x R
: 9
: 9
7=< 8
7 8
>
>yx
Zrqtszc;\
c
7
l T >@?
8
8
l ?
/ o
cw?nm
T >@?
? /
7 :
9
o
cw?nm
c
c
7 qts
>yx
*
} *|{ l
T >@?
8
cw?nm
?
L L
GH
>yx
{
J H HI T >@?
8
/
KML
9;: qts
o
N
c"?
9;:
}
V W >
W W
qts
OP x P PQ T
8
~
>@? J
x
Y ?
VXW >
W W
OP P PQ
q)
8
}
T >@? J
~ ?
Y
8
~
For we have which is the magnitude of the electric field due to an infinite sheet of charge. 1
?v
Due to a Disk of Charge (cont’d) dr r
dE
P
x R
8
: 9
V W >
W W
qts
OP x P PQ T > J
~
?
Y ?
V W W W
8
W
: 9
OP W P
qts
W P
x P P P
P PQ
J
W J
W e
T
a J
8
GH
qts
H
OP H
P
P
HI
For
PQ J
W / W W
KML L L L L L L ?
W
P H
~
V W ?
T
H x H
>
a
~ :
9
Y
{
> J
? Y
N
we can use the binomial expansion
_
8 T Z
\ /
?
T
J
J
J
q J
?
q J
8
9 qts
~
: GH
OP H
P
P HI
H
J P
J
V W M K L W L W L W L W L
? T
H x
PQ J
q
>@? Y N
J
Due to a Disk of Charge (cont’d) dr r
P
x R
8
OP qts
P x
9
:
W W W
J P PQ
`
W
?
T P
V W ~
: 9
J
q J
~ ?
s
>@?
x
8
9
>@? Y
:
>@?
From far away the disk looks like a point charge!
dE
Electric Field Lines The electric field vector E is tangent to the electric field line at each point.
The number of electric field lines per unit area through a surface that is perpendicular to the lines is proportional to the magnitude of the electric filed in that region.
Homework Set 13 - Due Fri. Apr. 30
Read Section 19.6
Answer Questions 19.10 & 19.11
Do Problems 19.16, 19.17, 19.18 & 19.20
