Điện trường hệ vật dẫn quả cầu – bản phẳng

Hình 1: Phân bố cường độ điện trường (colormap jet)
Hình 2: Phân bố cường độ điện trường (colormap hot)
Hình 3: Vector cường độ điện trường và các đường đẳng thế
Hình 4: Vector cường độ điện trường và các đường đẳng thế (colormap jet)

Code chương trình Matlab

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function E_Field_Plate_Sphere
% Author: Tran Hai Cat
% Lecturer in Physics, HCM University of Technology and Education
% - Dai hoc Su pham Ky thuat Tp. Ho Chi Minh
% Created: 2019.08.01
clc;
clear variables;
close all;

%% INPUT DATA FOR USER:
L = 2; % plate length
R = 5e-2; % sphere radius
a = 0.5; % 2a - distance between plates
Vmax = 100;
Vmin = -100;

Nx = 100;
Ny = 50;

eps = (Vmax-Vmin)/1e5;

contour_range_V = linspace(Vmin,Vmax,41);
xmin = -2; xmax = 2; ymin = -0.5; ymax = 1.5;

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

mpx = ceil(Nx/2); % Mid-point of x
mpy = ceil(Ny/4); % 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));

% Length of plate in terms of number of grids:
length_plate = floor(Nx*L/(xmax-xmin));
lp = floor(length_plate/2);
% Position of plate on y axis
position_sphere = floor(Ny*a/(ymax-ymin));
pp1 = mpy;
pp2 = mpy+position_sphere;

% Initializing Sphere Potentials:
phi = linspace(0,2*pi,120);
for i = 1:length(phi)
x_sphere = R*cos(phi(i));
px_sphere = ceil(Nx*(x_sphere-xmin)/(xmax-xmin));
y_sphere = a+R*sin(phi(i));
py_sphere = ceil(Ny*(y_sphere-ymin)/(ymax-ymin));

V(px_sphere,py_sphere) = Vmax;
V_const(px_sphere,py_sphere) = 1;
end

% Initializing Plate Potentials:
V(mpx-lp:mpx+lp,pp1) = Vmin;
V(mpx,pp2) = Vmax;
V_const(mpx-lp:mpx+lp,pp1) = 1;
V_const(mpx,pp2) = 1;

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-Vmin);
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');
contour(x,y,V,contour_range_V,'linewidth',0.5);
hold on,
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

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