Question

Exercise: Consider the configuration of conductors and potentials shown in Figure 14.7. Write a MATLAB program that solves Laplace equation in the area bounded by the conductors. Plot the contours of the voltage and the lines of the electric field Gap Gap 50N Gap 50 V 2 c 2 cm 100 v 0 V o V 4 cm 0 V 4 cm Figure 14.7 The configuration of the exercise of Set 14.

Write the MatLab code and solve by hand as well.

Answer 1

Laplace’s equation is a partial differential equation and its solution relies on the boundary conditions imposed on the system, from which the electric potential is the solution for the area of interest. Analytic solutions to this equation can be found using the method of separation of variables .Often, even Laplace’s equation is too difficult or unruly to solve analytically, and computational methods must be used

This is the required MATLAB code

This program calculates the solutian to Laplace 's equationwhich is %defined as the divergence of the gradient of a certian potential (or %steady state heat flow ). The program calculates the solutions to Laplace's %equation using the relaxation method , and does so for a square grid that %has a 100 unit principal length. clear all; close all: cle x 0:1:100; %defines the physical space for the model y 0:1:100; %(creates the grid to plot solution values on) V(101,101)-0; % Setall points to zero. for i-1:101 1ey (1,1)-10; % set one boundary to a cert ain function (in Volts). end for i-1:101 ( 1, i) # 0; % Set one boundary to a certain function (in Volts). end for i=1:101 (1,101) = o; % set one boundary to a certain function (in Volts). end for 1-1:101 V(101,1)-Oi % Set one boundary to to certain function (in Volts). end %sets the number of iterations. needs to be quite high to get %good representation of the solution n 1000: for ia: 1:n; %for loop using the relaxation method to calculate the potential for ix# 2:100; % all internal points for iy# 2:100; %(N-2 total internal points for each dimension) V(ty.ix) (Wiy-1,1x) + w(1у +1 , їх) + V(ty, iz-1) + V(sy.īx+1))/4; end end end Y; % Print out values for the electric potential (uncomment if you really % want to see this, but it !» 1000 numbers) [X,Y] meshgrid (x,y); % plots mesh of the solution .. subplot (2,2,1); surf(x,Y,v) %plots a surface plot of the solution xlabel(x); ylabely ) xlabel( 'Potential in Volts) subplot (2,2,2);contour (X.Y,Vlahows a (shaded) contour plot of the solution (potentia contour(V) xlabelx ylabel'yiclabel (potential) subplot (2,2,3); eontour3(X,Y,V) %shows as contour plot with colored lines xlabel(; ylabely) alabel( Potential in Volts') subplot (2,2,4): meshe (X,Y,V) %shown mesh plot of the xolution xlabel(i ylabel('y' alabel 'Potential in Volt