Capacitance of hollow sphere Q a Capacitance of

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Capacitance of hollow sphere Q a

Capacitance of hollow sphere Q a

Capacitance of parallel plates

Capacitance of parallel plates

Capacitance of a circular disk r Voltage at origin = 0 a R 2

Capacitance of a circular disk r Voltage at origin = 0 a R 2

Capacitance of a circular disk Charge density is uniform s 0 r a R

Capacitance of a circular disk Charge density is uniform s 0 r a R 2

Capacitance of a circular disk Charge density decreases s 0(1 r/a) r a R

Capacitance of a circular disk Charge density decreases s 0(1 r/a) r a R 2

Method of moments ° a known charge causes a potential ° what is the

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 ° Ph. D theses, faculty publications

V(a) V(b) Q 1 Q 2

V(a) V(b) Q 1 Q 2

V(a) V(b) Q 1 Q 2

V(a) V(b) Q 1 Q 2

>> A = [1/4 1/5 ; 1/5 1/4] ; >> V = [1 ;

>> A = [1/4 1/5 ; 1/5 1/4] ; >> V = [1 ; 1] ; >> Q = V’ / A % ‘transpose Q= 2. 2222 >> QQ = A V; QQ = 2. 2222

complications? ? ? r 1 r 2 The potential @ each charge sphere is

complications? ? ? r 1 r 2 The potential @ each charge sphere is specified as V 1 or V 2 Locations are with respect to an origin.

solution to singularity Assume that the potential is uniform within the sphere and it

solution to singularity Assume that the potential is uniform within the sphere and it is equal to the value at its edge r = a.

1 1 1

1 1 1

>Q = > > >> 4. 69 2. 56 4. 69

>Q = > > >> 4. 69 2. 56 4. 69

Capacitor C = Q / V rj a ri a a b b a

Capacitor C = Q / V rj a ri a a b b a

Capacitor C = Q / Harrington – p. 27 b V b Dwight 200.

Capacitor C = Q / Harrington – p. 27 b V b Dwight 200. 01 & 731. 2 b b

Capacitor C = Q / b V b b b

Capacitor C = Q / b V b b b

MATLAB > > clear; clf; N=9; az=37. 5; el=5;

MATLAB > > clear; clf; N=9; az=37. 5; el=5;

%identify subareas > for m = 1: 2 > for h = 1: N

%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

> for h=1: 2*N*N > for k=1: 2*N*N > aa=norm(a(h, : ) a(k, :

> 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 %calculate matrix elements

%set voltages on plates > for h=1: N*N > V(h) = +1/2; > V(h+N*N)

%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:

%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:

%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); end QT

QT=0; for j=1: N*N QT=QT+Q(j); end QT

Capacitance of a circular disk Charge density is uniform s 0 r a R

Capacitance of a circular disk Charge density is uniform s 0 r a R 2

Capacitance of a circular disk Charge density is noniform r a R 2

Capacitance of a circular disk Charge density is noniform r a R 2

C = 3. 9420 C = 3. 2367 C = 2. 7026 C =

C = 3. 9420 C = 3. 2367 C = 2. 7026 C = 3. 6332

Hwang & Mascogni – “Electrical capacitance of a unit cube” – Journal of Applied

Hwang & Mascogni – “Electrical capacitance of a unit cube” – Journal of Applied Physics 3798 3802 (2004).

2 a C a

2 a C a

a Sava V. Savov February 1, 2004 h

a Sava V. Savov February 1, 2004 h

 • E = s /2 • V = x s /2 x. •

• E = s /2 • V = x s /2 x. • V 1 = 2 [0/2 s 1 + 1 s 2 + 2/2 s 3 ] • V 2 = 2 [1/2 s 1 + 0 s 2 + 1/2 s 3] • V 3 = 2 [2/2 s 1 + 1 s 2 + 0/2 s 3] 1 2 3

 • V 1 0 • V 2 = 2 1/2 • V 3

• V 1 0 • V 2 = 2 1/2 • V 3 2/2 • pn junction • double layer • VLSI 1 0 1 2/2 s 1 1/2 s 2 0 s 3 1 2 3

pn junction • • • V[ 2 d] V[0] V[d] V[2 d] = linear

pn junction • • • V[ 2 d] V[0] V[d] V[2 d] = linear 0. 36 0. 18 0 + 0. 18 + 0. 36 quadratic 2 3/2 or 0 + 3/2 +2

matrix • • • 0 1/2 2/2 3/2 42 1 0 1 2 3

matrix • • • 0 1/2 2/2 3/2 42 1 0 1 2 3 2 1 0 1 4/2 3/2 2/2 1/2 0

MATLAB • [V] = [matrix] [Q] V V x x

MATLAB • [V] = [matrix] [Q] V V x x

Calculate vector “V” • • clear clf m=input('What is the size of the matrix?

Calculate vector “V” • • clear clf m=input('What is the size of the matrix? . . . '); for i=1: m v(i)=Na. N; end v

Calculate matrix “a” • for i=1: m • for j=1: m • a(i, j)=Na.

Calculate matrix “a” • for i=1: m • for j=1: m • a(i, j)=Na. N; • end

 • for i=1: m • for j=i: m • if i==j • a(i,

• 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

 • for i=2: m • for j=1: b(i) 1 • if j==1 •

• 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 • a

Calculate and plot • • • q=v/a; q'; plot(q(jmin: jmax)) hold on plot(v(jmin: jmax))

Calculate and plot • • • q=v/a; q'; plot(q(jmin: jmax)) hold on plot(v(jmin: jmax))

V z

V z