ΠE∆IA
B
17/02/2016
Θµα 1
(α)
λ
ln
Φ(x, y, z) =
4π0
(
)
(x + a)2 + (y − b)2 (x − a)2 + (y + b)2
[(x − a)2 + (y − b)2 ] [(x + a)2 + (y + b)2 ]
x, y ≥ 0.
Potential Φ (x,y)
4
3.5
3
y
2.5
2
1.5
1
0.5
0
0
1
2
x
3
4
Figure 1: Potential distribution for a = b = 1 m and λ/0 = 1 V.
(β)
~ =
E
λ (x − a)îx + (y − b)îy
λ (x + a)îx + (y − b)îy
−
+
2
2
2π0 (x − a) + (y − b)
2π0 (x + a)2 + (y − b)2
λ (x + a)îx + (y + b)îy
λ (x − a)îx + (y + b)îy
−
2
2
2π0 (x + a) + (y + b)
2π0 (x − a)2 + (y + b)2
(γ)
σ(x = 0, y ≥ 0) =
λa
−1
1
+
π a2 + (y − b)2 a2 + (y + b)2
1
Red=Field Lines, Blue=Equal Potential −− λ = 1, a = 1, b = 1
4
3.5
3
y
2.5
2
1.5
1
0.5
0
0
1
2
x
3
4
Figure 2: Electric field lines (red) and equipotential lines (blue) for a = b = 1 m and λ/0 = 1 V.
0
Charge Density, σ(y)
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
0
0.5
1
1.5
2
y
2.5
3
3.5
4
Figure 3: Surface charge density at x = 0 grounded plane, for a = b = 1 m and λ/0 = 1 V.
2
Θµα 2
(α)
Φ1 (z) =
Φ2 (z) =
P0 2 1
P0
z +
V0 −
d z,
20 d
`
20
1
P0
P0
V0 −
d z+
d,
`
20
20
0 ≤ z ≤ d,
d ≤ z ≤ `.
(β)
~ 1 = − P0 z + 1 V0 − P0 d îz ,
0 ≤ z ≤ d,
E
d `
2
0
0
~ 2 = − 1 V0 − P0 d îz ,
E
d ≤ z ≤ `,
`
20
~ 1 = − 0 V0 − P0 d îz ,
D
0 ≤ z ≤ d,
`
20
~ 2 = − 0 V0 − P0 d îz ,
D
d ≤ z ≤ `.
`
20
(γ)
ρb = −
P0
,
d
σb (z = 0+ ) = 0,
σb (z = d− ) = P0 ,
0
P0
V0 −
d ,
`
20
0
P0
σ(z = `+ ) = +
V0 −
d .
`
20
σ(z = 0− ) = −
(δ)
Lw
P0 d 1
C = C(V0 ) = 0
1−
,
`
20 V0
3
Normalized Capacitance, C/C0
1.5
1
0.5
0
0
0.5
1
1.5
Normalized Voltage, V0/VF
2
2.5
Figure 4: Normalized capacitance C/C0 (C0 = 0 Lw/`) as a function of the normalized voltage
V0 /VF (VF = P0 d/20 ).
4
Θµα 3
µ0 a2
L12 (Θ) = L21 (Θ) =
4
(
"
b cos Θ
h sin Θ + (c/2)
h sin Θ − (c/2)
−
2
2
2
1/2
h cos Θ + (b/2) [h2 + (c/2)2 + (b/2)2 + hc sin Θ]
[h2 + (c/2)2 + (b/2)2 − hc sin Θ]1/2
(c b/2) cos Θ
+
[h2 + (c/2)2 + hc sin Θ] [h2 + (b/2)2 + (c/2)2 + hc sin Θ]1/2
+
(c b/2) cos Θ
[h2 + (c/2)2 − hc sin Θ] [h2 + (b/2)2 + (c/2)2 − hc sin Θ]1/2
5
)
#
Θµα 4
(α)
Φ(x, y) =
Φ(x, y) =
∞
X
n=1
∞
X
An ekn x cos(kn y),
(x < 0),
An e−kn x cos(kn y),
(x > 0),
n=1
1 2σaΛ
π
sin n
,
2
2
1 + 2 n π
2
2π
= n .
Λ
An =
kn
(β)
~ =
E
~ =
E
(
−
∞
X
kn x
An kn e
)
cos(kn y) îx +
n=1
(∞
X
)
An kn e−kn x cos(kny) îx +
n=1
(
∞
X
kn x
An kn e
)
(x < 0),
)
(x < 0).
sin(kn y) îy
n=1
(∞
X
An kn e−kn x sin(kn y) îy
n=1
(γ)
Φ(x = 0, y) =
∞
X
An cos(kny).
n=1
Potential Φ (x,y)
0.5
0.4
0.3
0.2
y
0.1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
−0.4
−0.2
0
x
0.2
0.4
0.6
Figure 5: Sample electric potential visualization for one period for Λ = 1,m, σa Λ/0 = 1.
6
Red=Field Lines, Blue=Equal Potential −− L = 1, σ /ε = 1
0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0
.
−0
0
x
−0.5
−0.6
0.5
101
−0.4
−0.2
0
x
0.2
32
0.0010132
−0
.0010132
32
−0.5
−0.5
0.0
0.0
2
0.0010132 1013−0.0010132
−00.0010132
.010132
−0.0010132
6
32
39−
01
30 0.07
.01
.0
−0
09
0
−
25
01
−0.4
6
9
03
03
0.
01
−0.4
2
.
−0
−0.3
10
13
6
39
30
.0
−0 61
6
50
.0
−0.3
0.
0
1
−0.2
.0010132
−0
0.0010132
−0
−0.2
2
13
10
0.0
61
06
05
0. 396
0
03
0.
−0.1
32
−0.0010132
0.0010132
066
y
01
−0.1
5
92
70
.0
0
39 −
6
010132
−0.
−0.0010132
0.0010132
0.0101
0.0 32
30
39
6
0.070925
03
0
.01
−0
.0
10
13
2
0.4
−0
.0
50
6
−0 61
.0
30
39
0.4
0 .0 5
y
0.5
6
Equipotential Lines
a 0
0.5
0.4
0.6
Figure 6: Equipotential lines (blue) and electric field lines (red) for one period Λ = 1,m, σa Λ/0 = 1.
L = 1, σ /ε = 1
a 0
Normalized Potential at x=0, Φ(y)/Φ0
0.1
0.08
0.06
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−2
−1.5
−1
−0.5
0
0.5
x coordinate
1
1.5
2
Figure 7: Normalized electric potential Φ/Φ0 (Φ0 = σaΛ/0 ) at x = 0 for four periods for Λ = 1,m,
σa Λ/0 = 1.
7