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

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

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