1
Lecture 6
Electric fields in dielectrics and
conductors.
Electric field lines.
EXAMPLE: Ring
Below is a ring of radius
R
with uniform linear charge
distribution
λ
. Find the electric field along the
z
axis.
x
P
y
z
x
z
d
θ
d
q
=
λ
Rd
θ
r
=
+
2
2
R
z
P
y
2
dq
dE
k
r
=
G
2
2
Rd
k
R
z
λ
θ
=
+
cos
z
dE
dE
ϕ
=
G
2
2
2
2
Rd
z
k
R
z
R
z
λ
θ
=
+
+
x
P
z
Cylindrical
symmetry: only
E
z
matters
φ
zRd
λ
θ
y
(
)
3/2
2
2
k
R
z
=
+
(
)
2
, net
3/2
0
2
2
z
zR
E
k
d
R
z
π
λ
θ
=
+
∫
(
)
3/2
2
2
2
zR
k
R
z
πλ
=
+
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2
From the ring to the charged disk…
Disk of radius
R
with uniform surface charge density
σ
can
be thought of as made of rings of radii
r
= 0 to
r
=
R
and
linear charge density
σ
dr.
(
)
πσ
=
∫
disk
3/2
2
R
R
zr
E
k
dr
(
)
πσ
⎡
⎤
⎢
⎥
= −
⎢
⎥
1/2
1
2
R
k
z
+
0
2
2
r
z
+
⎣
⎦
2
2
0
r
z
(
)
πσ
⎛
⎞
⎜
⎟
= −
−
⎜
⎟
+
⎝
⎠
1/2
2
2
1
1
2
k
z
z
r
z
(
)
πσ
⎛
⎞
⎜
⎟
=
−
⎜
⎟
+
⎝
⎠
1/2
2
2
2
1
z
k
R
z
(
)
σ
ε
⎛
⎞
⎜
⎟
=
−
⎜
⎟
+
⎝
⎠
1/2
2
2
0
1
2
z
R
z
… and to the infinite plane
Take the limit
R
→
∞
(
)
σ
σ
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 Spring '07
 Johnson
 Physics, Charge, Electric Fields, Electric charge, dθ, Eext

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