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from Electric Field of a Parallel Plate capacitor using 2D Poisson Equation by Praveen Ranganath
Electric potential and E-field of a cross section of a parallel plate capacitor is calculated.

Parallel_plate_2D.m
%-------------------------------------------------------------------------%
%  This simple program computes the Electric Fields due to 
%  Parallel plate Capacitors using the Finite difference method (FDM)  
%-------------------------------------------------------------------------%

clc
close all; clear all;

%-------------------------------------------------------------------------%
%                   SYMBOLS USED IN THIS CODE                             
%-------------------------------------------------------------------------%

% E = Total electric field matrix using Poisson's equation
% V = Potential matrix
% Nx = Number of grid points in X- direction
% Ny = Number of grid points in Y-Direction
%-------------------------------------------------------------------------%


%-------------------------------------------------------------------------%
%                         INITIALIZATION                                  
%          Here, all the grid, size, charges, etc. are defined
%-------------------------------------------------------------------------%

% Enter the dimensions

Nx = 101;     % Number of X-grids
Ny = 101;     % Number of Y-grids
mpx = ceil(Nx/2); % Mid-point of x
mpy = ceil(Ny/2); % Mid point of y
   

Ni = 750;  % Number of iterations for the Poisson solver

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

T = 0;            % Top-wall potential
B = 0;            % Bottom-wall potential
L = 0;            % Left-wall potential
R = 0;            % Right-wall potential

%-------------------------------------------------------------------------%
% Initializing edges potentials
%-------------------------------------------------------------------------%

V(1,:) = L;
V(Nx,:) = R;
V(:,1) = B;
V(:,Ny) = T;

%-------------------------------------------------------------------------%
% 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_plate = 51;  % Length of plate in terms of number of grids  
lp = floor(length_plate/2);

position_plate = 15; % Position of plate on x axis
pp1 = mpx+position_plate;
pp2 = mpx-position_plate;

for z = 1:Ni    % Number of iterations
        
        for i=2:Nx-1
        for j=2:Ny-1      
            
            % The next two lines are meant to force the matrix to hold the 
            % potential values for all iterations
            
                V(pp1,mpy-lp:mpy+lp) = 100;
                V(pp2,mpy-lp:mpy+lp) = -100;
                
                V(i,j)=0.25*(V(i+1,j)+V(i-1,j)+V(i,j+1)+V(i,j-1));
        end
        end
        
end

% Take transpose for proper x-y orientation
V = V';

[Ex,Ey]=gradient(V);
Ex = -Ex;
Ey = -Ey;



% Electric field Magnitude
 E = sqrt(Ex.^2+Ey.^2);  

x = (1:Nx)-mpx;
y = (1:Ny)-mpy;

% Contour Display for electric potential
figure(1)
contour_range_V = -101:0.5:101;
contour(x,y,V,contour_range_V,'linewidth',0.5);
axis([min(x) max(x) min(y) max(y)]);
colorbar('location','eastoutside','fontsize',14);
xlabel('x-axis in meters','fontsize',14);
ylabel('y-axis in meters','fontsize',14);
title('Electric Potential distribution, V(x,y) in volts','fontsize',14);
h1=gca;
set(h1,'fontsize',14);
fh1 = figure(1); 
set(fh1, 'color', 'white')


% Contour Display for electric field
figure(2)
contour_range_E = -20:0.05:20;
contour(x,y,E,contour_range_E,'linewidth',0.5);
axis([min(x) max(x) min(y) max(y)]);
colorbar('location','eastoutside','fontsize',14);
xlabel('x-axis in meters','fontsize',14);
ylabel('y-axis in meters','fontsize',14);
title('Electric field distribution, E (x,y) in V/m','fontsize',14);
h2=gca;
set(h2,'fontsize',14);
fh2 = figure(2); 
set(fh2, 'color', 'white')

% Quiver Display for electric field Lines
figure(3)
contour(x,y,E,'linewidth',0.5);
hold on, quiver(x,y,Ex,Ey,2)
title('Electric field Lines, E (x,y) in V/m','fontsize',14);
axis([min(x) max(x) min(y) max(y)]);
colorbar('location','eastoutside','fontsize',14);
xlabel('x-axis in meters','fontsize',14);
ylabel('y-axis in meters','fontsize',14);
h3=gca;
set(h3,'fontsize',14);
fh3 = figure(3); 
set(fh3, 'color', 'white')


%-------------------------------------------------------------------------%
% REFERENCE
%           SADIKU, ELEMENTS OF ELECTROMAGNETICS, 4TH EDITION, OXFORD
%-------------------------------------------------------------------------%

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