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PHY 5346
HW Set 4 Solutions – Kimel
3. 2.10 As done in class we simulate the electric field E0 by two charges at ∞
σ = 30E0 cosθ
a) This charge distribution simulates the given system for cosθ > 0.We have treated this probem
in class. The potential is given by
φx⃗ = −E0 1 − a
3
r3
r cosθ
Using
σ = −0 ∂∂n
φ|surface
We have the charge density on the plate to be
σplate = −0 ∂∂z
φ|z=0 = 0E0 1 − a
3
ρ3
For purposes of plotting, consider
σplate
0E0
= 1 − 1
x3
0
0.2
0.4
0.6
0.8
1
2
4
6
8
10
x
σboss = 30E0 cosθ =
For plotting, we use σboss
30E0
= cosθ
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
1.2
1.4
b)
q = 30E02πa2 ∫
0
1
xdx = 3π 0E0a2
c) Now we have
φr⃗ =
1
4π 0
q
r⃗ − d⃗
+
q ′
r⃗ − d⃗ ′
−
q
r⃗ + d⃗
−
q ′
r⃗ + d⃗ ′
where q ′ = −q a
d
, d ′ = a
2
d
.
σ = −0 ∂∂r
φ|r=a =
−q
4π
d2 − a2
a a⃗ − d⃗
3 −
d2 − a2
a a⃗ + d⃗
3
q ind = 2πa2
−q
4π ∫0
1
d2 − a2
a a⃗ − d⃗
3 −
d2 − a2
a a⃗ + d⃗
3
dx
q ind =
−qa2d2 − a2
2a
1
da
1
d − a −
1
a2 + d2
+
1
d + a
−
1
a2 + d2
q ind = −12 q
d2 − a2
d
2d
d2 − a2
−
2
a2 + d2
= −q 1 −
d2 − a2
d a2 + d2
HW Set 4 Solutions – Kimel
3. 2.10 As done in class we simulate the electric field E0 by two charges at ∞
σ = 30E0 cosθ
a) This charge distribution simulates the given system for cosθ > 0.We have treated this probem
in class. The potential is given by
φx⃗ = −E0 1 − a
3
r3
r cosθ
Using
σ = −0 ∂∂n
φ|surface
We have the charge density on the plate to be
σplate = −0 ∂∂z
φ|z=0 = 0E0 1 − a
3
ρ3
For purposes of plotting, consider
σplate
0E0
= 1 − 1
x3
0
0.2
0.4
0.6
0.8
1
2
4
6
8
10
x
σboss = 30E0 cosθ =
For plotting, we use σboss
30E0
= cosθ
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
1.2
1.4
b)
q = 30E02πa2 ∫
0
1
xdx = 3π 0E0a2
c) Now we have
φr⃗ =
1
4π 0
q
r⃗ − d⃗
+
q ′
r⃗ − d⃗ ′
−
q
r⃗ + d⃗
−
q ′
r⃗ + d⃗ ′
where q ′ = −q a
d
, d ′ = a
2
d
.
σ = −0 ∂∂r
φ|r=a =
−q
4π
d2 − a2
a a⃗ − d⃗
3 −
d2 − a2
a a⃗ + d⃗
3
q ind = 2πa2
−q
4π ∫0
1
d2 − a2
a a⃗ − d⃗
3 −
d2 − a2
a a⃗ + d⃗
3
dx
q ind =
−qa2d2 − a2
2a
1
da
1
d − a −
1
a2 + d2
+
1
d + a
−
1
a2 + d2
q ind = −12 q
d2 − a2
d
2d
d2 − a2
−
2
a2 + d2
= −q 1 −
d2 − a2
d a2 + d2