electrostatics numerical integration
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Electrostatics
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FE ru
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Coulomb
Newtons
meter
volts
electric field
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r
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dydE
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Voltage -- work
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Voltage – work
Superposition
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k = 2
j = 3
k = 3
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a
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A
dz
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
+
a
x
z
yb
A
dz
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
+
a
x
z
yb
A
dz
#1
for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on endendxlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black')
#1
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
+
a
x
z
yb
A
dz
#1
#2;
for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on endendxlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black')
#1
#2
for w=-1:2 for ddz=1:10; dz=ddz*10^w; V=0; loglog(dz,V,'o') hold on endendxlabel('dz','fontsize',18) ylabel('V','fontsize',18) set(gca,'fontsize',18) whitebg('black')
10-1
100
101
102
103
10-3
10-2
10-1
100
dz
V
slope = -1
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.
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.”
% electric field at different distancesclear;clffor 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);endloglog(1: 100, coefficient,'-s')hold onplot([10/(10^(1/2)) 100], [1000 1],'--','linewidth', 3)xlabel ('z/a','fontsize', 18)ylabel ('coefficient','fontsize', 18)set(gca,'fontsize', 18)grid onlegend ('numerical integration','slope = -2', 3)
Comparison of the two integration techniques
12
x
y
z
12
12
6
V(z = 6) = ?
Find V(z)
66
12
6
x
y
z
V(z = 6) = ?
Find V(z)
