Capacitance of hollow sphere Q a Capacitance of

Capacitance of hollow sphere Q a

Capacitance of parallel plates

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 2

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 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

>> 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 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 is equal to the value at its edge r = a.

1 1 1

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

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

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

Capacitor C = Q / b V b b b

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

%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, : )); > 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) = 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); end QT

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

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

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

2 a C a

a Sava V. Savov February 1, 2004 h

• 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 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 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 2 1 0 1 4/2 3/2 2/2 1/2 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)=Na. N; end v

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, 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 • 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))

V z