Cecilia first drops off the edge mostly moving parallel the x coordinate. As she gets near bottom of the parabola, she starts moving turning toward

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4 Cecilia first drops off the edge mostly moving parallel the x coordinate. As she gets near bottom of the parabola, she starts moving turning toward walking nearly parallel to the y-axis

5 Electric Dipole Figure 1. We see the positive 'pole' for the positively charged point charge at x = +1, as well as the negative pole corresponding to the negative piont charge at x = -1 Figure 2. Dipole Electric Field. Can see the classic point out of + charge, points into - charge pattern. Note that a slice of x values was removed to avoid the singulatirites that occur at x = +/-1.

6 Starting at (x,y,z) = 7,2,1, We see that the pressure field will push the dandelion down the hill (decreasing values of x and y). Also note that the pressure is higher nearer to the ground, so the dandelion seed will be pushed toward the sky. This is suggested also in the vector field plot for grad P.

7 Cecilia Crab Walk: Matlab Code %% example code for generating vector field plots % Matlab draws vector fields using the 'quiver' function. % in this example, we use Cecilia the crab. We found the negative of the gradient is given % by: % -grad f = -x/2 ihat + 1/10 jhat %she starts at the point (4,10,3), so let's smartly choose our x %and y domains to see her path startint at (xo, yo) = 4,10. %set up the grid and define function values there xvals = linspace(-4.5, 4.5, 25); %this defines a list of values -5 to 5 in steps of 0.05 yvals = linspace(9, 12, 25); %choose limits sensibly based on how large a function becomes along that coordinate [x,y] = meshgrid(xvals, yvals); %this makes a 2d grid of points out of the x and y values specified % elevation as function of x and y z = (x.^2)/4 - y/10; %gradient field u = -x/2; v = 1/10*ones(size(y)); %% make figure to show lake bed surface elevation figure; surf(x,y,z); colormap jet xlabel('x') ylabel('y') title('cecilia Crab Walk') set(gca, 'fontsize', 16) %plot the starting point hold on xo = 4; yo = 10; zo = 3; plot3(xo, yo, zo, 'ko', 'markerfacecolor', 'k') %% make figure for gradient field figure; %use autoscalefactor to make quiver arrows bigger, easier to view quiver(x,y,u,v, 'linewidth', 1.5, 'AutoScaleFactor', 2);

8 %plot cecilia's initial position hold on; xo = 4; yo = 10; plot(xo, yo, 'ko', 'markerfacecolor', 'r', 'markersize', 7) xlabel('x') ylabel('y') title('cecilia Crab Walk') set(gca, 'fontsize', 16)

9 Electric Dipole: matlab code % Electric dipole: +q placed at +xo, -q placed at -xo %set up the grid and define function values there xvals = [-1.5:0.05:-1.1, -0.9:0.05:0.9, 1.1:0.05:1.5]; yvals = [-1.5:0.05:-0.02, 0.02:0.05:1.5]; % xvals = linspace(-1.5, 1.5, 50); %this defines a list of values -5 to 5 in steps of 0.05 % yvals = linspace(-1.5, 1.5, 50); %choose limits sensibly based on how large a function becomes along that coordinate [x,y] = meshgrid(xvals, yvals); %this makes a 2d grid of points out of the x and y values specified % let k = 1. in reality it is 1/(4*pi*eps0), changes absolute scale, but % not the overall shape of the voltage surface % let xo = 1. Again, we could place them at say xo = +/- 10, but this only % changes the absolute scale xo = 1; k = 1; % electric potential as function of x and y Voltage = k./( (x-xo).^2 + y.^2).^0.5 - k./( (x+xo).^2 + y.^2).^0.5; %gradient field u = k*(x-xo)./( (x-xo).^2 + y.^2).^1.5 - k*(x+xo)./( (x+xo).^2 + y.^2).^1.5; v = k*y./( (x-xo).^2 + y.^2).^1.5 - k*y./( (x+xo).^2 + y.^2).^1.5; %% Note matlab can compute gradients. This is very useful if you are working with experimental data % and don't have a closed-form ready solution to type in. [Fx, Fy ] = gradient(voltage); % >> doc gradient %% make figure to show lake bed surface elevation figure; HC = surf(x,y,voltage); HC.EdgeColor ='none'; HC.FaceAlpha = 0.75; colormap jet xlabel('x') ylabel('y') zlabel('v(x,y)') title('dipole Electric Potential') set(gca, 'fontsize', 16) %% make figure for gradient field figure; %use autoscalefactor to make quiver arrows bigger, easier to view quiver(x,y,u, v, 'linewidth', 1.5, 'AutoScaleFactor', 1);

10 xlabel('x') ylabel('y') title('dipole Electric Field') set(gca, 'fontsize', 16) axis equal

11 Pressure Field Problem: matlab code %% Matlab workshop prob 6: Under Pressure % Matlab draws vector fields using the 'quiver' function. %In this example we are considering a pressure varying P(x,y,z): % P = (x-3).^(1/2).* (y-5).^2.* exp(-z); %we'll plot the pressure at ground surface z = 0; % THis makes the e^-z term = 1. %set up the grid and define function values there xvals = linspace(4, 10, 20); %this defines a list of values -5 to 5 in steps of 0.05 yvals = linspace(0, 4, 20); %choose limits sensibly based on how large a function becomes along that coordinate zvals = linspace(0,2,5); [x,y] = meshgrid(xvals, yvals); %this makes a 2d grid of points out of the x and y values specified zo = 0; %plotting at ground level only so we can visualize the surface P = (x-3).^(1/2).* (y-5).^2; %% make figure to show lake bed surface elevation figure; HC = surf(x,y,p); HC.EdgeColor = 'none'; colormap jet xlabel('x') ylabel('y') zlabel('pressure') title('pressure field over (x,y,0)') set(gca, 'fontsize', 16) %plot the starting point hold on xo = 7; yo = 2; zo = 1; plot3(xo, yo, zo, 'ko', 'markerfacecolor', 'k') %% Plot the gradient field % now add in z to the grid (lattice) to plot wind field [x,y, z] = meshgrid(xvals, yvals, zvals); %this makes a 2d grid of points out of the x and y values specified %-grad(p) u = -0.5*(x-3).^(-1/2).*(y-5).^2.*exp(-z); v = -2* (x-3).^(-1/2).* (y-5).*exp(-z);

12 w = -(x-3).^(-1/2).* (y-5).^2.* -exp(-z); % make figure for gradient field figure; %use autoscalefactor to make quiver arrows bigger, easier to view quiver3(x,y,z, u,v, w, 'linewidth', 1.5, 'AutoScaleFactor', 3); %plot pressure gradient = direction of wind flow hold on; plot3(xo, yo, zo, 'ko', 'markerfacecolor', 'r', 'markersize', 7) xlabel('x') ylabel('y') zlabel('z'); title('wind flow field') set(gca, 'fontsize', 16)

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