EEE161 Applied Electromagnetics Laboratory 3

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1 Dr. Milica Marković Applied Electromagnetics Laboratory page 1 EEE161 Applied Electromagnetics Laboratory 3 Instructor: Dr. Milica Marković Office: Riverside Hall milica@csus.edu Web: milica 1 Visualizing Scalar Fields in Matlab To visualize scalar fields in Matlab we can use the following functions: slice, contourslice, patch, isonormals, camlight and lightning. Please note that more detailed explanation about the these organization functions shown here can be found in Matlab help. 1.1 slice Slice is a command that shows the magnitude of a scalar fields on a plane that slices the volume where the potential field is visualized. The format of this command is as shown below. slice(x,y,z,v,xslice,yslice,zslice) Where X, Y, and Z are coordinate of points where the scaler function is calculated, the are the values of the scaler function at those points, and the last three vectors xslice, yslice, and zslice life are showing where will the volume will be sliced. X flies has three points at which the x-axis will be slice. They are -1.2,.8, 2. This means that the volume will be slice with a plane that is perpendicular to x-axis and it crosses the x-axis at points -1.2,.8 and 2. An example of slice command is given below. in the example below there is an additional command colormap, that s collars the volume with a specific pallette. To see more about different color maps, see Matlab help. clc clear all [x,y,z] = meshgrid(-2:.2:2,-2:.25:2,-2:.16:2); v = x.*exp(-x.^2-y.^2-z.^2); xslice = [-1.2,.8,2]; yslice = 1; zslice = [-2,0]; slice(x,y,z,v,xslice,yslice,zslice) colormap hsv 1.2 contourslice Contourslice command will display equipotential lines on a plane being the volume where the potential field is visualized. An example of contourslice function is shown below.

2 Dr. Milica Marković Applied Electromagnetics Laboratory page 2 [x,y,z] = meshgrid(-2:.2:2,-2:.25:2,-2:.16:2); v = x.*exp(-x.^2-y.^2-z.^2); % Create volume data [xi,yi,zi] = sphere; % Plane to contour contourslice(x,y,z,v,xi,yi,zi) view(3) 1.3 patch Patch command creates a patch of color. 1.4 isonormals Command isonormals creates equipotential surfaces. 1.5 camlight camlight( headlight ) creates a light at the camera position.camlight( right ) creates a ligh right and up from camera. camlight( left ) creates a light left and up from camera.camlight with no arguments is the same as camlight( right ). camlight(az,el) creates a light at the specified azimuth (az) and elevation (el) with respect to the camera position. The camera target is the center of rotation and az and el are in degrees. 1.6 lighting lighting flat selects flat lighting. lighting gouraud selects gouraud lighting. lighting phong selects phong lighting. lighting none turns off lighting.

3 Dr. Milica Marković Applied Electromagnetics Laboratory page 3 2 Gradient, Divergence and Curl 2.1 Del Operator Del operator, defined in Equation 2.1 is used to define gradient V, divergence A, laplacian 2 V, and curl A of a field. = x a x + y a y + z a z (1) 2.2 Gradient Gradient is a derivative for functions of several variables. Function of two variables x and y is shown in Figure 1 (a). We can imagine that this is a temperature distribution in a Silicon slab. Gradient of a this temperature distribution gives the maximum rate of change of the temperature. It shows both the direction and magnitude of the rate of change, as in Figure 1 (b). As we see, gradient operates on a scalar function (temperature) and the result of the gradient operation is a vector (the direction and magnitude of temperature change). Vector s direction points in the direction of the maximum increase or decrease of temperature. Vector s magnitude shows the magnitude of temperature change. Gradient is mathematically defined as T = T T T x + y + z (2) x y z Sometimes, the rate of change of in a specified direction a l needs to be found. In Figures 1 (c)-(d) rate of change of temperature is shown in a x and a y directions. This vector is called directional derivative, and can be found by finding the component of the gradient vector in a certian direction, as shown in Equation 3. Compare the vector field in Figure 1 (b) with Figures 1 (c)-(d). Fields in Figures 1 (c)-(d) are the a x and a y components of the field in Figure 1 (b). dt dl = T a l (3) Syntax of Gradient in Matlab v = -2:0.2:2; [x,y] = meshgrid(v); z = x.* exp(-x.^2 - y.^2); [px,py] = gradient(z,.2,.2); vec=[px,py] a=zeros(21)+1 b=zeros(21)

4 Dr. Milica Marković Applied Electromagnetics Laboratory page 4 (a) Function Graph (b) Gradient and Countours of Function (c) Directional Derivative in a x direction. (d) Directional Derivative in a y direction. Figure 1: Function z = xe x2 y 2. c=zeros(21) d=zeros(21)+1 vec1=[a,b] sol=a.*px+b.*py sol2=c.*px+d.*py %contour(v,v,z), hold on, quiver(v,v,px,py) %hold off %figure, mesh(x,y,z) %figure, quiver(x,y,sol,b) %figure, quiver(x,y,c,sol2) 2.3 Divergence Divergence is flux through a closed surface. Divergence operates on a vector function and the result of the divergence operation is a scalar. Result of divergence operation on a vector function, therefore has only the magnitude. Divergence is really a flux of a vector through the closed surface. To

5 Dr. Milica Marković Applied Electromagnetics Laboratory page 5 understand flux better, if we visualize the field intensity as field line density per unit area, as shown in Figure 2 then we can simply count the number of field lines per unit area to find the magnitude of flux. It matters which way the field lines go through the surface. If the same number of lines go into and out of a closed surface, this means that there is no source or sink inside the closed surface, and the field is called divergenceless. If the net flux through a closed surface is positive (with respect to the normal on the surface oriented outwards), then there is a source of the field inside the closed surface. If the flux is negative, a sink is inside the closed surface. Divergence is mathematically defined as A = A x + A y + A z (4) Figure 2: Vector representation of Nort-West wind of 10mph. (a) Vector field (b) Divergence (Flux) of Vector Field Figure 3: Vector Field A = x 3 a x + y 3 a y + z 3 a z. The field is divergenceless if it s divergence is zero. Examples of Divergenceless fields are given in Figure.

6 Dr. Milica Marković Applied Electromagnetics Laboratory page 6 Figure 4: Divergenceless Vector Field A = 0.2 a x Syntax of Divergence in Matlab clc clear all [x,y,z]=meshgrid(-2:0.2:2) x1=0 y1=0 z1=0 %cc=[-0.05,0.05] %quiver3(x,y,z,x,y,z) div = divergence(x,y,z,x.^3,y.^3,z.^3); xslice=[-1.2,0.8,2]; yslice=[2,0,1]; zslice=[-2,0.2]; slice(x,y,z,div,xslice,yslice,zslice) shading flat; %lims=volumebounds(x,y,z,cc); %axis(lims); daspect([1 1 1]); camlight; 2.4 Curl Curl describes the rotation (curling) of the field. Curl operates on a vector function and the result of the curl operation is a vector. Result of curl operation on a vector function, therefore has a magnitude and direction. Curl describes the curling (rotation) of the field at a point. If the field is curling, then the direction of the curl will be perpendicular to the curling and the magnitude will describe the amount of rotation. Curl of a uniform field is zero and the field is called irrotational. Curl is mathematically defined as

7 Dr. Milica Marković Applied Electromagnetics Laboratory page 7 Or A = a x a y a z x y z A x A y A z A = ( Az y A ) ( y Ax a x + z z A ) ( z Ay a y + x x A ) x a z (5) y Curl can be estimated by using the paddle in a vector field. For example, in Figure 5, a paddle is inserted in the vector field A = x a x. Since the field is stronger with increasing x, those filed lines will be pushing the paddle stronger then the ones closer to (0,0). Therefore, the paddle will be rotating counter-clockwise looking from the top. Direction of curl is then found by using the right hand rule. The curl vector is always perpendicular to the field at a point. Figure 5: Paddle Inserted in Vector Field Syntax of Curl in Matlab clc clear all [X Y Z]= meshgrid(-3:1.5:3); s = size(x); V= zeros(s); A = -Y; B = X; [CURLX2, CURLY2, CURLZ2] = curl(a,b,v) subplot(2,1,1), quiver3(x,y,z,a,b,v,1); subplot(2,1,2), quiver3(x,y,z,curlx2,curly2,curlz2,1); 3 Visualize the Gradient of the scalar field T = x 2 + y 2 + z 2 (6)

8 Dr. Milica Marković Applied Electromagnetics Laboratory page 8 4 Calculate and Plot the Divergence and Curl of the vector field A = x 2 a x + y 2 a y + z 2 a z (7) 5 Find the source 1. Divergence of a magnetic field is equal to zero. Write a Matlab code to check if the vector fields given below could be magnetic fields. 2. The source of the magnetic field is current. If magnetic field is known, curl of the magnetic field is equal to the current density at that point in space. Write Matlab code to check if the current exists at that point in space. Two fields are given: B = y x (8) B = ρ φ (9) Write the Matlab code to find curl for both fields. Comment on the value of the curl. Look at the field graphs in problem 3. Imagine inserting a paddle in the field at certain points. Will the paddle turn clockwise or counterclockwise? Compare the paddle rotation to the direction of the curl. 6 Plot the Electric Field Defined by Electrostatic potential. The electric field is a negative gradient of the electrostatic potential. If the function of electrostatic potential is given V (R) = 1, first plot the electrostatic potential, then find the electric field and R plot it using gradient function. Write the Matlab code to plot the electric field (using quiver3) in 3 Dimensions. In order to do a 3-D plot, you will first have to convert the equation from polar to Cartesian coordinates on paper. The following conversions are relevant: 1. Convert Spehrical to Cartesian Coordinates on paper (find what is R as a function of x,y,z.). 2. Write potenital in Cartesian Coordinates. 3. Specify all points of interest in Matlab using meshgrid 4. Calculate potential in Matlab (write the equation for potential in cartesian coordinates) 5. Use the gradient function in Matlab to find the electric field.[ex,ey, EZ] = gradient(-v). Note that the gradient returns three components of the field. 6. To plot the electric field use quiver3 and to plot potential use Plot function and just plot the V as a function of R (2-D graph).

9 Dr. Milica Marković Applied Electromagnetics Laboratory page 9 Conclusion Solve all of the above examples by hand and compare the results with the ones you found during the lab. In addition, for each of the problems, write what you have learned. Do not write an essay, just in a few sentences write what you have done in each problem, and specifically what you have learned from the work done. Due Date Submit the printout of the lab work and the conclusion by NEXT FRIDAY. LAB IS DUE next Friday in the lab. No late labs.