 # post lab. Anonymous
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Laboratory #4 EE Science II Laboratory #04 Charge Distributions and Electric Field Visualization in 3D Laboratory Assignment The purpose of this lab is to perform various calculations related to electric fields, e.g. charge distributions and particle movement inside electric fields. This lab is linked to the special lecture in the Advanced Visualization Center (AVC). You will also need to refer to the MATLAB tutorial posted on Canvas for some fundamental aspects of the activities presented below. Part I: Theory - Warm Up Answer the following questions 1. If you bring a negative charge (Q2) near to a positive charge (Q1) what happens to the orientation of the resultant total electric field? Also, what happens to the electric field direction when a positive charge is near Q1? 2. A continuous charge density function describes how a certain amount of charge is spread out over some region. Depending on the distribution of the electric charges in a region of space, we have three different “categories” of density functions - linear, surface and volumetric charge density. Provide brief descriptions of these categories in your own words. 3. In a uniform charge distribution, the charge density has the same value at every point within the region. However, in a non-uniform distribution, the density does not have a fixed value - it varies as a function of position. Answer the following: a) Assume that you know the total charge and total volume occupied by a non-uniform charge distribution. Can you calculate the charge density at any point in the object? Explain Part II: Electric Field of Point Charges The magnitude of an electric field represents the Coulomb force per unit of charge on any electric charge within the field. The convention sets out the direction of the electric field vector towards the direction of movement of an infinitesimal positive charge immersed in the electric field. The following figures are some examples of electric field direction – the figure on the left is the electric field due to a dipole and the figure on the right is the electric field within an infinitely large parallel plate capacitor. University of South Florida 2 EE204-exp.docx Laboratory #4 The equation describing the electric field of a point charge is: 𝐸𝐸�⃗(𝑅𝑅) = 𝐸𝐸�⃗(𝑥𝑥,𝑦𝑦) = 𝑄𝑄 � 𝑄𝑄 𝑅𝑅� (𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑐𝑐𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑠𝑠) (4𝜋𝜋𝜋𝜋)𝑅𝑅2 1 1 𝑋𝑋 𝑌𝑌 �� � ∗ �𝑥𝑥� + 𝑦𝑦� � (𝑒𝑒𝑒𝑒𝑒𝑒𝑐𝑐𝑒𝑒𝑠𝑠𝑒𝑒𝑒𝑒𝑐𝑐 𝑒𝑒𝑐𝑐𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑐𝑐𝑒𝑒𝑠𝑠 𝑒𝑒𝑐𝑐 𝑧𝑧 = 0) 2 (4𝜋𝜋𝜋𝜋𝑟𝑟 𝜋𝜋0 ) �√𝑋𝑋 2 + 𝑌𝑌 2 � √𝑋𝑋 2 + 𝑌𝑌 2 √𝑋𝑋 2 + 𝑌𝑌 2 1 Then, using �𝑘𝑘 = (4𝜋𝜋𝜖𝜖 𝑟𝑟 𝜖𝜖0 ) � you can express 𝐸𝐸�⃗(𝑥𝑥,𝑦𝑦) as: �⃗(𝐱𝐱,𝐲𝐲) = �𝐱𝐱�𝐄𝐄𝐱𝐱 + 𝐲𝐲�𝐄𝐄𝐲𝐲 � = �𝐱𝐱� 𝐄𝐄 𝐐𝐐∗𝐤𝐤∗𝐗𝐗 ��𝐗𝐗 𝟐𝟐 +𝐘𝐘 𝟐𝟐 � 𝟑𝟑 + 𝐲𝐲� 𝐐𝐐∗𝐤𝐤∗𝐘𝐘 𝟑𝟑 ��𝐗𝐗 𝟐𝟐 +𝐘𝐘 𝟐𝟐 � 𝐐𝐐∗𝐤𝐤∗𝐗𝐗 � = �𝐱𝐱� �𝐑𝐑 (𝐱𝐱,𝐲𝐲) � 𝟑𝟑 + 𝐲𝐲� 𝐐𝐐∗𝐤𝐤∗𝐘𝐘 𝟑𝟑 � (1) �𝐑𝐑 (𝐱𝐱,𝐲𝐲) � The activities in this section are designed to give you practice in visualizing electric fields and the use of the field concept to solve problems. You are asked to use MATLAB to compute the electric field produced by the particle A. As necessary you should refer to the lecture notes and the MATLAB tutorial posted on Canvas. 1. The Problem - Read the situation described in the following paragraph. Suppose a charged particle, A is present at (x, y) = (ax, ay) in free space. The electric charge of this particle has a value of Q. You are required to compute the electric field produced by the particle A. Assume that no other fields are present in this region of space. A brief representation is shown in the following image. Table 1- Equations describing the position vector 𝑃𝑃�⃗ for particle A and its electric charge value Q Z X ��⃗ 𝑷𝑷(ax,ay) Y ��⃗ 𝑷𝑷(𝒂𝒂𝒂𝒂, 𝒂𝒂𝒂𝒂) [ (𝑤𝑤 + 2) , (𝑤𝑤 − 1) ] Q (𝑤𝑤)(10−3 ) (W = Group number - It will be assigned by the TA) Q 2. Compute the position vector and electric charge value - open a spreadsheet or use a calculator to calculate Q and the position of A in space by using the equations in Table 1 and complete Table 2 below. Note: to complete this step you need to substitute your assigned group number in for w in the equations. University of South Florida 3 EE204-exp.docx Laboratory #4 Table 2 – Particle A position and charge value P (ax, ay) [ , ] Q 3. Validate the results using MATLAB - Create a new script in the MATLAB script editor. Copy the template code below into MATLAB’s script editor. Make sure to type the equations / values for P (ax, ay), Q, e0, and er. Save the script using the name “EE204_your_last_names_II” and press the run button. Fill in Table 3 and compare against Table 2. %-------------------------------------------------------------------------% % This script computes the Electric Fields due to point charge % in a 2-D plane using the Coulomb's Law %-------------------------------------------------------------------------% clear; clc; %-------------------------------------------------------------------------% % SYMBOLS USED IN THIS CODE %-------------------------------------------------------------------------% % W = Group number (Assigned by TA) % P = [ax,ay] = coordinates for the position of particle A % e0 = free space permittivity % er = Relative permittivity % Q = charge value is stored here % k = (Coulomb's constant)*(er) % Nx = Number of grid points in X-Direction % Ny = Number of grid points in Y-Direction % L = Distance coverture % X,Y = coordinate system % R = distance between a selected point and the location of charge % Ex = X-Component of Electric-Field % Ey = Y-Component of Electric-Field %-------------------------------------------------------------------------% % INITIALIZATION OF INPUT PARAMETERS % Here, all the input constants are defined %-------------------------------------------------------------------------% W = 1; %Use your group number here. Remember to modify before running the code! P = [,] %complete the equation for position vector e0 = 8.854*10^-12 er = 1 Q = %complete the equation for Charge value L = 15; Nx = 15; Ny = 15; k = 1/(4*pi*e0*er) Table 3 – Particle A: position and charge value (MATLAB) P(ax, ay) Q 4. Generate the Meshgrid – append the next template to your script. %-------------------------------------------------------------------------% % X Y SYSTEM CALCULATION %-------------------------------------------------------------------------% x=-L:2*L/(Nx-1):L; y=-L:2*L/(Ny-1):L; [X,Y]= meshgrid(x,y); R = sqrt(((X-P(1)).^2)+((Y-P(2)).^2)); %Calculating the distance from source University of South Florida 4 EE204-exp.docx Laboratory #4 What is the purpose of the above code lines? Hint: Look for plotting functions in the MATLAB Tutorial 5. Compute the electric field - append the next template to your script. Complete the code by entering the parameters for the quiver function. Run the code and change the title of the figure to display your group number and last names. Save your figure and submit it with your postlab report. %-------------------------------------------------------------------------% % CALCULATION OF THE x AND y COMPONENTS OF THE ELECTRIC FIELD %-------------------------------------------------------------------------% Ex = (Q*k.*(X-P(1)))./(R.^3); Ey = (Q*k.*(Y-P(2)))./(R.^3); %-------------------------------------------------------------------------% % PLOTTING THE ELECTRIC FIELD %-------------------------------------------------------------------------% % The following lines are intended to normalize and visualize the electric field % -------------------------------------Ex1 = Ex.*R.^1.5; Ey1 = Ey.*R.^1.5; figure contourf(X,Y,1./R.^0.5,10) %Plotting equipotential lines title('Change the title to your group number and last names') xlabel('x') ylabel('y') colormap(jet) colorbar hold on % ----------------End of field normalization --------------------q = quiver(X,Y, , ); %complete the parameters for quiver function q.Color='w'; % defining the color of the electric field arrows hold off Observations: The arrows in your MATLAB plot represent the electric field due to the charge and the colored contours represent equipotential surfaces around the charge. Comment on the direction of the electric field. How does the strength of the field vary with respect to distance? What are equipotential surfaces? From theory, what will be the shape of equipotential surfaces around a single charge? Do you see this in your MATLAB plot? Why/why not? Obtain your TA’s signature here after completing Part II on the first page Part III: Electric Field of a Dipole 1. Read the situation described below Two charged particles, A and B are separated by a distance d in free space. Particles A and B have an electric charge of Q1 and Q2 respectively. Compute the net electric field due to these two charged particles by assuming a coordinate system centered at the midpoint of the line connecting A and B. A brief representation is shown in the following image. University of South Florida 5 EE204-exp.docx Laboratory #4 Table 4- Equations describing the charge and position of A and B Z (−1)𝑊𝑊 (𝑊𝑊 ∗ 10−3 ) Q1 (Particle A Charge) Q2 (Particle B Charge) −(𝑊𝑊)(1.5 ∗ 10−2 ) d 10(𝑊𝑊) (W = Group number assigned by the TA) ��⃗ 𝑩𝑩(bx,by) X ��⃗(ax,ay) 𝑨𝑨 Y 2. Compute the position vector and charge values - open a spreadsheet or use a calculator to calculate Q1, Q2, and their positions. Complete Table 5 below using the equations described in Table 4. Table 5 – Parameters describing the charge and position for particle A and particle B Q1 Q2 𝒅𝒅 - distance between A and B ��⃗(x,y) - position of particle A 𝑨𝑨 [ d/2 , 0 ] = [ , 0 ] ��⃗ 𝑩𝑩(x,y) - position of particle B [ -d/2 , 0 ] = [ , 0 ] 3. Validate the results using MATLAB - Create a new script in the MATLAB script editor and copy the template below. Make sure to type the equations for the values of Q1, Q2, d, A and B. Save the script using the name “EE204_your_last_names_III” and press the run button. Fill Table 6 and compare against Table 5. %-------------------------------------------------------------------------% % This simple program computes the Electric Fields due to dipole % in a 2-D plane using the Coulomb's Law %-------------------------------------------------------------------------% clear; clc; %-------------------------------------------------------------------------% % SYMBOLS USED IN THIS CODE %-------------------------------------------------------------------------% % W = Group number (Assigned by TA) % e0 = free space permittivity % er = Relative permittivity % Q1 = charge I value is stored here % Q2 = charge II value is stored here % Pi = ? % k = (Coulomb's constant)*(er) % Nx = Number of grid points in X-Direction % Ny = Number of grid points in Y-Direction % L = Distance coverture % X,Y = coordinate system % ax,ay = coordinates for the position of Particle A % bx,by = coordinates for the position of particle B % Ra = distance between a selected point and the location of particle A % Rb = distance between a selected point and the location of particle B % Ea = [Eax,Eay] Electric-Field produced by Particle A % Eb = [Ebx,Eby] Electric-Field produced by Particle B %-------------------------------------------------------------------------% % INITIALIZATION OF INPUT PARAMETERS % Here, all the input constants are defined %-------------------------------------------------------------------------% W = 3; %Use your group number here. Remember to modify before running the code! e0 = 8.85*10^-12; er = 1; Q1 = ; %complete the equation Q2 = ; %complete the equation d = 10*W; A = [ ,0]; %complete the equation for x component of position vector (Particle A) University of South Florida 6 EE204-exp.docx Laboratory #4 B = [ ,0]; %complete the equation for x component of position vector (Particle B) k = 1/(4*pi*e0*er); Table 6 – Computed values for charge and position of particles A and B (MATLAB) Q1 Q2 𝐷𝐷 - distance between A and B A(x,y) - position of particle A B(x,y) - position of particle B 4. Plot the electric field - append the next template to your script. Make sure to code the lines for Rb, Eby and the quiver function parameters. Run the code and change the title of the figure to your group number and last names. Save a copy of the figure. %-------------------------------------------------------------------------% % X Y SYSTEM CALCULATION %-------------------------------------------------------------------------% L = 50; Nx = 20; Ny = 20; x=-L:2*L/(Nx-1):L; y=-L:2*L/(Ny-1):L; [X,Y]= meshgrid(x,y); Ra = sqrt(((X-A(1)).^2)+((Y-A(2)).^2)); Rb = ; %complete the equation to calculate the distance for B %-------------------------------------------------------------------------% % CALCULATION OF THE x AND y COMPONENTS OF THE ELECTRIC FIELD %-------------------------------------------------------------------------% Eax = (Q1*k.*(X-A(1)))./(Ra.^3); Eay = (Q1*k.*(Y-A(2)))./(Ra.^3); Ebx = (Q2*k.*(X-B(1)))./(Rb.^3); Eby = ; %complete the equation to calculate Y component of B field %-------------------------------------------------------------------------% % PLOTTING THE ELECTRIC FIELD %-------------------------------------------------------------------------% % The following lines are intended to normalize and visualize the electric field % -------------------------------------Eax1 = Eax.*Ra.^1.5; Eay1 = Eay.*Ra.^1.5; Ebx1 = Ebx.*Rb.^1.5; Eby1 = Eby.*Rb.^1.5; figure contourf(X,Y,(1./Ra.^0.5)+(1./Rb.^0.5),10); colormap(jet); colorbar; title('Change this title to your group number and last names') xlabel('x') ylabel('y') hold on % ----------------End of field normalization --------------------q = quiver(X,Y , + , + ); %complete (Apply superposition on normalized fields) q.Color='w'; hold off Observations: The arrows in your MATLAB plot represent the electric field due to the charges and the colored contours represent equipotential surfaces. Identify A and B in the MATLAB figure. Comment on the field distribution and the equipotential surfaces. University of South Florida 7 EE204-exp.docx Laboratory #4 Obtain your TA’s signature here after completing Part III on the first page Part IV: Linear Charge distribution Sometimes, the distances between charges are much smaller than the space from the charges to some point of interest. Under such situations, the system of electric charges is called continuous. That is, the system is equivalent to a total electric charge distribution along a line, surface or volume. The procedure to develop the equations describing this scenario for a line charge is as follow: • We start by considering the electric charge distribution as a set of small elements, each of which contains a small charge ΔQ. Here the [X, Y] is the position vector of the evaluation point and [𝑥𝑥0 , 𝑦𝑦0 ] is the position of the 𝐝𝐝𝐐𝐐 element of the rod. The electric field produced by a differential portion of the rod is as follow 𝐝𝐝𝐐𝐐 ∗ 𝐤𝐤 ∗ (𝐗𝐗 − 𝐱𝐱𝐱𝐱) �⃗(𝐱𝐱,𝐲𝐲) = �𝐱𝐱�𝐝𝐝𝐄𝐄𝐱𝐱 + 𝐲𝐲�𝐝𝐝𝐄𝐄𝐲𝐲 � = �𝐱𝐱� 𝚫𝚫𝐄𝐄 ��(𝐗𝐗 − 𝐱𝐱𝐱𝐱)𝟐𝟐 + (𝐘𝐘 − 𝐲𝐲𝐱𝐱)𝟐𝟐 � • + 𝐲𝐲� 𝐝𝐝𝐐𝐐 ∗ 𝐤𝐤 ∗ (𝐘𝐘 − 𝐲𝐲𝐱𝐱) ��(𝐗𝐗 − 𝐱𝐱𝐱𝐱)𝟐𝟐 + (𝐘𝐘 − 𝐲𝐲𝐱𝐱)𝟐𝟐 � 𝟑𝟑 � Next, we substitute the differential charge element by the following equation 𝐝𝐝𝐐𝐐 = 𝐝𝐝𝐱𝐱 ∗ 𝛌𝛌, where dx is a infinitesimal portion of the rod, λ is the lineal charge density and 𝑐𝑐𝑄𝑄 is the electric charge of the infinitesimal portion of the rod. �⃗(𝐱𝐱,𝐲𝐲) = �𝐱𝐱� 𝚫𝚫𝐄𝐄 𝛌𝛌 ∗ 𝚫𝚫𝚫𝚫 ∗ 𝐤𝐤 ∗ (𝐗𝐗 − 𝐱𝐱𝐱𝐱) ��(𝐗𝐗 − 𝐱𝐱𝐱𝐱)𝟐𝟐 + (𝐘𝐘 − 𝐲𝐲𝐱𝐱)𝟐𝟐 � • 𝟑𝟑 𝟑𝟑 + 𝐲𝐲� 𝛌𝛌 ∗ 𝚫𝚫𝚫𝚫 ∗ 𝐤𝐤 ∗ (𝐘𝐘 − 𝐲𝐲𝐱𝐱) ��(𝐗𝐗 − 𝐱𝐱𝐱𝐱)𝟐𝟐 + (𝐘𝐘 − 𝐲𝐲𝐱𝐱)𝟐𝟐 � 𝟑𝟑 � Finally, we evaluate the total field at P(x,y,z) due to the charge distribution. Adding the contributions of all the infinitesimal charge elements (that is, by applying the superposition principle - integration). 𝒍𝒍/𝟐𝟐 �⃗(𝐱𝐱,𝐲𝐲) = �𝐱𝐱�𝐄𝐄𝐱𝐱 + 𝐲𝐲�𝐄𝐄𝐲𝐲 � = �𝐱𝐱� ∫𝒍𝒍/𝟐𝟐 𝐄𝐄 𝐝𝐝𝐄𝐄𝐱𝐱 + 𝐲𝐲� ∫𝒂𝒂𝐱𝐱=−𝒍𝒍/𝟐𝟐 𝐝𝐝𝐄𝐄𝐲𝐲� (2) 𝒂𝒂𝐱𝐱=−𝒍𝒍/𝟐𝟐 1. Read the situation described below Consider a charged cylinder, C with a length l and negligible width (i.e., approximate the charged cylinder as a line charge distribution with charge density of value 𝑒𝑒ℎ𝑐𝑐). Table 7- Equations describing the charge density and length of the charged cylinder 𝒓𝒓𝒓𝒓𝒓𝒓 - Charge density (C/m) (𝑤𝑤)(10−9 ) l – length of the cylinder (𝑤𝑤 + 15) (W = Group number assigned by the TA) Use MATLAB to calculate the total electric field produced by the charged cylinder C by following the steps below: University of South Florida 8 EE204-exp.docx Laboratory #4 2. Compute the charge density and length values - open a spreadsheet or use a calculator to calculate rho and l. Complete Table 8 below using the equations in Table 7. Table 8 - Parameters describing the charge density and length of C 𝑒𝑒ℎ𝑐𝑐 l 3. Validate the results using MATLAB - Create a new script in the script editor and copy the template below. Make sure to type the equations for rho and l. Save the script using the name “EE204_your_last_names_IV” and press the run button. Fill Table 9 and compare against Table 8. %-------------------------------------------------------------------------% % This program computes the Electric Fields due to a charged rod % in a 2-D plane using the Coulomb's Law %-------------------------------------------------------------------------% clear; clc; %-------------------------------------------------------------------------% % SYMBOLS USED IN THIS CODE %-------------------------------------------------------------------------% % W = Group number (Assigned by TA) % rho = Charge density [C/m] % l = length of C % e0 = free space permittivity % er = Relative permittivity % d = charge density for linear rod % X,Y = coordinate system % R = distance between a selected point and the location of infinitesimal charge % dE = Electric-Field due to infinitesimal charge element % E = Electric-Field % x0,y0 = coordinates for the location of infinitesimal charge (Integration variable) % Pi = ? % k = (Coulomb's constant)*(er) % Nx = Number of grid points in X-Direction % Ny = Number of grid points in Y-Direction % L = Distance coverture %-------------------------------------------------------------------------% % INITIALIZATION OF INPUT PARAMETERS ...
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