method of moments

1. method of moments

2. This technique was developed in a Ph.D. thesis by Ken Mei in 1962. His adviser was Jean van Bladel.

3. 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. V(a) V(b) Q2 Q1 2 equations 2 unknowns

6. V(a) V(b) Q2 Q1

7. MATLAB!!

8. >> A = [1/4 1/5 ; 1/5 1/4] ; >> V = [1 ; 1] ; >> Q = V’ / A % ‘transpose Q = 2.2222 2.2222 >> QQ = A \ V; or QQ = 2.2222 2.2222

9. complications??? r1 r2 • The potential @ each charge sphere is specified as V1 or V2 • Locations are with respect to an origin.

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

11. a = 1

12. 1 1 1 1 1 1

13. Q = • 4.69 2.56 4.69 -4.69 -2.56 -4.69 • >>

14. equal charge

15. Capacitor C = Q / V rj ri

16. MATLAB • clear; clf; • N=9; • az=37.5; • el=5;

17. %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

18. %calculate matrix elements • 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

19. %set voltages on plates • for h=1:N*N • V(h) = +1/2; • V(h+N*N) = -1/2; • end

20. %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);

21. %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)

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

23. What have you done? You have learned a technique, to accurately calculate the capacitance of a parallel plate capacitor! Big deal!

26. capacitance of a unit cube

27. capacitance of a unit cube

28. capacitance of a unit cube

29. 2a C a

31. note the charge density at the corner

32. a

33. Target shooting in Alaska

34. 65 N; 25 E Suomi Finland

35. Helsinki

36. Turku

37. Academic Kiuas-Seura is a sauna and sausage society, formed mainly by older members of Liisankadun Osakunnat, whose main task is taking sauna baths, sloping of time together and sensory testing of the sausages. The assembling is 6:30 on Tuesdays, the sausage tests continue in spite of the environmental terrorists.