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Electrostatics numerical integration

Electrostatics numerical integration. Electrostatics. +Q. +Q. +Q. - Q. electric field. r. y. r. y. . x. x. . . symmetry. r. y. . x. . . z. r. y. x. Gauss's law. infinite charged sheet. Voltage -- work. Voltage – work Superposition. numerical integration. y.

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Electrostatics numerical integration

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  1. Electrostatics numerical integration

  2. Electrostatics +Q +Q +Q - Q

  3. electric field

  4. r y r y  x x   symmetry

  5. r y  x  

  6. z r y x Gauss's law

  7. infinite charged sheet

  8. Voltage -- work

  9. Voltage – work Superposition

  10. numerical integration

  11. y b x + Dz z a

  12. y b x + Dz z a

  13. j = 3 y b j = 2 x j = 1 Rj,k k = 1 + k = 2 Dz z k = 3 a

  14. j 1 2 3 n-1 n 1 2 3 b k m-1 m a

  15. j = 3 y b j = 2 x j = 1 Rj,k k = 1 + k = 2 Dz z k = 3 a

  16. y b x + z a

  17. 5 13 6 8 4 3 12

  18. y b x + z a

  19. y b x A + dz z a

  20. clear; clf n=3; a=12; m=3; b=16; dz=12; V=0; for j=1: n-1 for k=1: m-1 A=[dz ((n/2)-j)*(a/(n-1)) ((m/2)-k)*(b/(m-1))]; R=norm(A); V=V+1/R; end end V V = .3077 = 4/13

  21. plot the result of the previous calculation

  22. clear n=3; a=12; m=3; b=16; dz=12; V=0; for j=1: n-1 for k=1: m-1 A=[dz ((n/2)-j)*(a/(n-1)) ((m/2)-k)*(b/(m-1))]; R=norm(A); V=V+1/R; end end V V = .3077 = 4/13 #1

  23. for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on end end xlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black') #1

  24. clear n=3; a=12; m=3; b=16; dz=12; V=0; for j=1: n-1 for k=1: m-1 A=[dz ((n/2)-j)*(a/(n-1)) ((m/2)-k)*(b/(m-1))]; R=norm(A); V=V+1/R; end end V V = .3077 = 4/13 #1 #2 ;

  25. for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on end end xlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black') #1 #2

  26. for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on end end xlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black')

  27. slope = -1

  28. Electric field normal to a surface • Two regions – • The first region is very close to the surface so the surface almost appears to be infinite in extent. • The second is at distances that are large with respect to the dimensions of the surface and the surface appears to be a point charge.

  29. quadrature function ”quad” • the function “quad” approximates the integral of a function from a to b with an error of 10- 6 using “recursive adaptive Simpson quadrature.” • This also holds true for “dblquad” & “triplequad.”

  30. % electric field at different distances clear;clf for z= 1: 100 f=inline('10*z./(sqrt(x.^2+y.^2+(z/10).^2).^3)'); coefficient(z) =dblquad(f, -.5, .5, -.5, .5, [ ],'',z); end loglog(1: 100, coefficient,'-s') hold on plot([10/(10^(1/2)) 100], [1000 1],'--','linewidth', 3) xlabel ('z/a','fontsize', 18) ylabel ('coefficient','fontsize', 18) set(gca,'fontsize', 18) grid on legend ('numerical integration','slope = -2', 3)

  31. Comparison of the two integration techniques

  32. 12 y V(z = 6) = ? Find V(z) 12 12 x 6 z

  33. 6 6 12 y V(z = 6) = ? Find V(z) 6 x z

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