Capacitance of hollow sphere a Q Capacitance of parallel plates.
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Transcript of Capacitance of hollow sphere a Q Capacitance of parallel plates.
V
QC
Capacitance of hollow sphere
a4QQ
aQ
a4
t
A
V
QC
Capacitance of parallel plates
Et
AS
t
A
S
S
t
A
Capacitance of a circular disk
ar
R 2 -
a0
S20 rdr
R4
),r(d),(V
a0
S rdrR4
)r(2)(V
)cos(r2rR 22
Voltage at origin = 0
a
0S rdr
r4
)r(2)0(V
a
0S dr2
)r()0(V
Capacitance of a circular disk
ar
R 2 -
a
0S dr2
)r()0(V Charge density is uniform s0
a
00S dr
2)0(V
a2
)0(V 0S
QC
V
a2C
2s0 a
V(0)
Capacitance of a circular disk
a
0S dr2
)r()0(V Charge density decreases s0(1-r/a)
S0a
0
r1 aV(0) dr2
S0V(0) a4
QC
V
2 a
0 0
4 rV(0) d 1 draaV(0)
4 a2 2a
4 a
ar
R 2 -
Method of moments° a known charge causes a potential ° what is the potential?° a measured or known potential is caused by an
unknown charge° what was that charge?° used in em to find current distributions on
antennas, scattering objects such as cubes, sleds, planes, and missles
° PhD theses, faculty publications
5
Q
4
Q
4
1aV 21
Q1Q2
V(a) V(b)
4
Q
5
Q
4
1bV 21
?Q?Q 21
5
Q
4
Q
4
1aV 21
Q1Q2
V(a) V(b)
4
Q
5
Q
4
1bV 21
2
1
Q
Q
4
1
5
15
1
4
1
4
1
)b(V
)a(V
4
1)b(V)a(Vlet
2
1
Q
Q
4
1
5
15
1
4
1
1
1
2
1
Q
Q
4
1
5
15
1
4
1
4
1
)b(V
)a(V
2
1
Q
Q
4
1
5
15
1
4
1
1
1
>> A = [1/4 1/5 ; 1/5 1/4] ;
>> V = [1 ; 1] ;
>> Q = V’ / A % ‘transposeQ =
2.2222 2.2222
>> QQ = A \ V;
QQ = 2.2222 2.2222
complications???
The potential @ each charge sphere is specified as V1 or V2
Locations are with respect to an origin.
r1 r2
4
1
V
V
2
1
2
1
Q
Q
22 rr
1
12 rr
1
12 rr
1
11 rr
1
solution to singularity
a4
Q)j,j(V.e.i
Assume that the potential is uniform within the sphere and it is equal to the value at its edge r = a.
4π
1
5
5
5
5
5
5
6
5
4
3
2
1
Q
Q
Q
Q
Q
Q1 29
141
12
14
15
1
1
1
1
1
1
>Q =
> 4.69 2.56 4.69 -4.69 -2.56 -4.69
>>>
a2
4s
a
020
s
r
rdrd
4
1)j,j(V
Aa s2
s
A
a
A24
s
a2
4s
a
020
s
r
rdrd
4
1)j,j(V
Aa s2
s
A
a
A24
s
Capacitor C = Q / V
ri
rj
-b-b
b
bN
a2b2
-a-a a
a
Capacitor C = Q / V
22
11
11
1dd
4
b
22
bb
bb
yx
1dydx
4
1)j,j(V
)8814.0(b2
21lnb2
-b-b
b
b
Harrington – p. 27 Dwight 200.01 & 731.2
b
y
b
x
Capacitor C = Q / V
2kj2
kj
2
yyxx
b2
4
1)k,j(V
-b-b
b
b
> clear; clf;
> N=9;
> az=37.5;
> el=5;
MATLAB
%identify subareas
> for m = 1:2> for h = 1:N> for k = 1:N> a(m - 1)*N*N +
(h - 1)*N + k, :)=[m, h, k];
> end> end> end
> for h=1:2*N*N> for k=1:2*N*N> aa=norm(a(h, :) - a(k, :));> if aa==0> b(h, k) = 2 * sqrt(pi);> else> b(h, k) =1 / aa;> end> end> end
%calculate matrix elements
%set voltages on plates
> for h=1:N*N
> V(h) = +1/2;
> V(h+N*N) = -1/2;
> end
%calculate charges
>Q = V*inv(b);
>%top plate
>QA(1:N*N) = Q(1:N*N);
>%bottom plate
>QB(1:N*N)=Q(N*N+1:2*N*N);
%plot> [x,y]=meshgrid(1:N);> for i=1:N> za(1:N,i)=QA((i-1)*N+1:(i-1)*N+N)';> zb(1:N,i)=QB((i-1)*N+1:(i-1)*N+N)';> end> mesh(x,y,za)> hold on> mesh(x,y,zb)
QT=0; for j=1:N*N QT=QT+Q(j);endQT
1N*1N
QT
117*117
3696.28
11.0
Capacitance of a circular disk
ar
R 2 -
a
0S dr2
)r()0(V
Charge density is uniform s0
a2C
C 7.88 a -- numerical
Capacitance of a circular disk
ar
R 2 -
a
0S dr2
)r()0(V
Charge density is noniform a88.7C 22
0ss
ra)r(
)0(V4
0s
a0 s rdr2)r(
V
1
V
QC
a8C
C = 3.9420
C = 3.2367
C = 2.7026
C = 3.6332
V6
QC sideone sideoneC6C
Hwang & Mascogni – “Electrical capacitance of a unit cube” – Journal of Applied Physics 3798-3802 (2004).
V
QC
aπε4QQ
C
aπε4C
+
+
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+
+
+
+
+
+
+
+
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2a
a
C
a
h
Sava V. Savov
February 1, 2004
h
aC
20
x.
1 2 3
• E = s /2• V = x s /2
• V2 = 2 [1/2 s1 + 0 s2 + 1/2 s3]
• V3 = 2 [2/2 s1 + 1 s2 + 0/2 s3]
• V1 = 2 [0/2 s1 + 1 s2 + 2/2 s3 ]
• V1 0 1 2/2 s1 • V2 = 2 1/2 0 1/2 s2 • V3 2/2 1 0
s3
1 2 3
•pn junction•double layer•VLSI
pn junction
• linear quadratic
• V[-2d] - 0.36 - 2
• V[-d] - 0.18 - 3/2
• V[0] = 0 or 0
• V[d] + 0.18 + 3/2
• V[2d] + 0.36 + 2
matrix
• 0 1 2 3 4/2
• 1/2 0 1 2 3/2
• 2/2 1 0 1 2/2
• 3/2 2 1 0 1/2
• 42 3 2 1 0
MATLAB
• [V] = [matrix] [Q]
V
V
x x
Calculate vector “V”
• clear
• clf
• m=input('What is the size of the matrix? ...');
• for i=1:m
• v(i)=NaN;
• end
• v
NaN
NaN
NaN
NaN
v
Calculate matrix “a”
• for i=1:m
• for j=1:m
• a(i,j)=NaN;
• end
• end
NaNNaNNaNNaN
NaNNaNNaNNaN
NaNNaNNaNNaN
NaNNaNNaNNaN
• for i=1:m• for j=i:m• if i==j• a(i,j)=0;• b(i)=i;• elseif i<j & j<m a(i,j)=(j-b(i)); elseif i<j & j==m• a(i,j)=(j-b(i))/2;• end• end• end
0NaNNaNNaN
NaN0NaNNaN
NaNNaN0NaN
NaNNaNNaN0
0NaNNaNNaN
NaN0NaNNaN
NaN10NaN
NaN210
0NaNNaNNaN
1/20NaNNaN
2/210NaN
3/2210
• for i=2:m• for j=1:b(i)-1• if j==1• a(i,j)=(b(i)-j)/2;• else• a(i,j)=b(i)-j;• end• end• end• a
0NaNNaN3/2
1/20NaN2/2
2/2101/2
3/2210
0123/2
1/2012/2
2/2101/2
3/2210
Calculate and plot
• q=v/a;
• q';
• plot(q(jmin:jmax))
• hold on
• plot(v(jmin:jmax))
V
z
)xtanh(V
2)xcosh(
1
dx
dVE
3)xcosh(
)xsinh(2
dx
dE