Điện trường vật dẫn có hình dạng phức tạp

Hình 1: Phân bố cường độ điện trường quanh vật dẫn (colormap jet)
Hình 2: Phân bố cường độ điện trường quanh vật dẫn (colormap hot)
Hình 3: Phân bố điện thế quanh vật dẫn

 

Code chương trình Matlab

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function E_Field_Complex_Body
% Author: Tran Hai Cat
% Lecturer in Physics, HCM University of Technology and Education
% - Dai hoc Su pham Ky thuat Tp. Ho Chi Minh
% Created: 2019.08.03
clc;
clear variables;
close all;

%% INPUT DATA FOR USER:
a = 0.5;
b = a/2;
Vmax = 100;

Nx = 500;
Ny = 250;

eps = Vmax/1e5;

contour_range_V = linspace(0,Vmax,41);

xmin = -2; xmax = 2; ymin = -1; ymax = 1;
%% CALCULATION
x = linspace(xmin,xmax,Nx);
y = linspace(ymin,ymax,Ny);

mpx = ceil(Nx/2); % Mid-point of x
mpy = ceil(Ny/2); % Mid point of y

hx = (xmax-xmin)/(Nx-1);
hy = (ymax-ymin)/(Ny-1);

A = 2/hx/hx+2/hy/hy;
B = 1/hx/hx;
C = 1/hy/hy;

V = zeros(Nx,Ny); % Potential (Voltage) matrix
V_const = zeros(Nx,Ny);

% Initializing edges potentials
V(1,:) = 0;
V(Nx,:) = 0;
V(:,1) = 0;
V(:,Ny) = 0;

% Initializing Corner potentials
V(1,1) = 0.5*(V(1,2)+V(2,1));
V(Nx,1) = 0.5*(V(Nx-1,1)+V(Nx,2));
V(1,Ny) = 0.5*(V(1,Ny-1)+V(2,Ny));
V(Nx,Ny) = 0.5*(V(Nx,Ny-1)+V(Nx-1,Ny));

% Initializing potentials:
ap = ceil(Nx*a/(xmax-xmin));
bp = ceil(Ny*b/(ymax-ymin));
V(mpx-ap:mpx+ap,mpy+bp) = Vmax;
V(mpx-ap:mpx+ap,mpy-bp) = Vmax;
V_const(mpx-ap:mpx+ap,mpy+bp) = 1;
V_const(mpx-ap:mpx+ap,mpy-bp) = 1;

tan_alpha = b/a;
x1 = linspace(-a,0,100);
y1 = b-(a+x1)*tan_alpha;
for i = 1:length(x1)
px = ceil(Nx*(x1(i)-xmin)/(xmax-xmin));
py = ceil(Ny*(y1(i)-ymin)/(ymax-ymin));

V(px,py) = Vmax;
V_const(px,py) = 1;
end
y1 = -b+(a+x1)*tan_alpha;
for i = 1:length(x1)
px = ceil(Nx*(x1(i)-xmin)/(xmax-xmin));
py = ceil(Ny*(y1(i)-ymin)/(ymax-ymin));

V(px,py) = Vmax;
V_const(px,py) = 1;
end
x1 = linspace(a,2*a,100);
y1 = (2*a-x1)*tan_alpha;
for i = 1:length(x1)
px = ceil(Nx*(x1(i)-xmin)/(xmax-xmin));
py = ceil(Ny*(y1(i)-ymin)/(ymax-ymin));

V(px,py) = Vmax;
V_const(px,py) = 1;
end
y1 = -(2*a-x1)*tan_alpha;
for i = 1:length(x1)
px = ceil(Nx*(x1(i)-xmin)/(xmax-xmin));
py = ceil(Ny*(y1(i)-ymin)/(ymax-ymin));

V(px,py) = Vmax;
V_const(px,py) = 1;
end

p = 1e100;
V_old = V;
while p>eps
for i=2:Nx-1
for j=2:Ny-1
if V_const(i,j)==0
V(i,j)=1/A*(B*(V(i+1,j)+V(i-1,j))+C*(V(i,j+1)+V(i,j-1)));
end
end
end
Delta_V = abs(V-V_old);
p = max(Delta_V(:));
V_old = V;
error = p/Vmax;

clc
fprintf('error=%f\n',error);
end

% Take transpose for proper x-y orientation
V = V';
[Ex,Ey]=gradient(V);
Ex = -Ex;
Ey = -Ey;
E = sqrt(Ex.^2+Ey.^2);
Emax = max(E(:));

%% FIGURES:
figure('name','Electric Field Magnitude','color','w','numbertitle','off');
hold on
pcolor(x,y,E)
axis image
shading interp;
colormap jet
colorbar('location','eastoutside','fontsize',14);
xlabel('x [m]','fontsize',14);
ylabel('y [m]','fontsize',14);
title('Electric Field Magnitude');
set(gca,'fontsize',14);
axis equal

figure('name','Electric Field Magnitude','color','w','numbertitle','off');
hold on
pcolor(x,y,E)
axis image
shading interp;
colormap hot
colorbar('location','eastoutside','fontsize',14);
xlabel('x [m]','fontsize',14);
ylabel('y [m]','fontsize',14);
title('Electric Field Magnitude');
set(gca,'fontsize',14);
axis equal

figure('name','Electric Field and Potential distribution','color','w','numbertitle','off');
pcolor(x,y,V)
axis image
shading interp;
colormap jet
hold on,
contour(x,y,V,contour_range_V,'linewidth',0.5,'linecolor','k');
quiver(x,y,Ex,Ey,2)
axis([min(x) max(x) min(y) max(y)]);
colorbar('location','eastoutside','fontsize',14);
xlabel('x [m]','fontsize',14);
ylabel('y [m]','fontsize',14);
title('Electric field and Potential distribution');
set(gca,'fontsize',14);
axis equal