clear all format short

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1 DESCRIPTION: This code plots electrostatic field lines for a collection of point charges. MAIN GOAL: To draw the field lines in such a way that their volume density is proportional to the electric flux penetrating any finite surface area. MAIN IDEA: Start field lines very close from a given charge. Then, if the initial points are distributed *uniformly* over a tiny sphere surrounding the charge, the propagated lines will automatically "adjust" themselves in space with correct volume density. MAIN PROBLEM: In general, it is impossible to *uniformly* distribute an arbitrary number M of points on the surface of a sphere. That can only be done for M = 4, 6, 8, 12, and 20, which correspond to the number of vertices of the so-called Platonic solids. NOTE 1: Thus, in our implementation of the code, the number of field lines originating on a given charge is proportional to the charge and equal to the number of vertices of the corresponding Platonic solid. Due to this restriction, only charges q = 1, 1.5, 2, 3, 5, -1, -1.5, -2, -3, -5, may be used. NOTE 2: If net charge is *positive*, only field lines originating on *positive* charges are propagated. In this case, propagation direction is *forward*. If net charge is *negative*, only field lines originating on *negative* charges are propagated. In this case, propagation direction is *backward*. NOTE 3: *Position vectors* should be entered as [1-3 5], or [1, -3, 5], etc. Make sure that the charges are "reasonably well" localized. AUTHOR: Andrei Galiautdinov Department of Physics & Astronomy University of Georgia Athens, GA ag@physast.uga.edu DATE: 2016 SPRING, February 08. clear all format short %% Welcome: fprintf(1,'* Welcome to PHYS4201/4202 fprintf(1,'* Electrostatic Field Lines Calculator! tocontinue = input('press Enter to Continue: '); %% Some technical details:

2 fprintf(1,'* Distance of initial line point from corresponding fprintf(1,'* charge, epsilon, epsilon = %%% Choose. Distance of initial point from a charge. fprintf(1,'* Number of iteration steps on each line, N_step, N_step = 80 %%% Choose fprintf(1,'* Size of each iteration step, step_factor, step_factor = %%% Choose %% A note on Platonic solids: fprintf(1,'* ATTENTION: fprintf(1,'* fprintf(1,'* The number of field lines originating on a given charge fprintf(1,'* is equal to the number of vertices of the corresponding fprintf(1,'* Platonic solid. fprintf(1,'* fprintf(1,'* fprintf(1,'* To find the coordinates of initial points of certain fprintf(1,'* field lines requires the knowledge of the Golden ratio: fprintf(1,'* fprintf(1,'* phi = (1+sqrt(5))/2 phi = (1+sqrt(5))/2 tocontinue = input('press Enter to Continue: '); %% Define positive charges: fprintf(1,'* Specify the NUMBER of positive charges: N_q_positive = input('enter N_q_positive NUMBER: ') fprintf(1,'* Available positive charge values: fprintf(1,'* 1, 1.5, 2, 3, 5 % tocontinue = input('press Enter to Continue: '); fprintf(1,'* Begin specifying positive charges and their positions... q_positive = zeros(1,n_q_positive); rq_positive = cell(1,n_q_positive); N_line_positive = zeros(1,n_q_positive) q_positive_net = 0; for n_q_positive = 1:N_q_positive

3 q_positive n_q_positive q_positive(n_q_positive) = input('enter CHARGE q_positive: ') rq_positive{1,n_q_positive} = input('enter rq_positive POSITION VECTOR: ') if q_positive(n_q_positive) == 1 fprintf(1,'* Charge: q = 1 fprintf(1,'* Number of field lines: 4 (Tetrahedron) N_line_positive(n_q_positive) = floor(4) %%% Notation: r_in_line{n_line,n_q} r_in_line_positive{1,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_positive{2,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_positive{3,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); r_in_line_positive{4,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); elseif q_positive(n_q_positive) == 1.5 fprintf(1,'* Charge: q = 1.5 fprintf(1,'* Number of field lines: 6 (Octahedron) N_line_positive(n_q_positive) = floor(6) r_in_line_positive{1,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 0, 0]/norm([1, 0, 0]); r_in_line_positive{2,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 0, 0]/norm([-1, 0, 0]); r_in_line_positive{3,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 1, 0]/norm([0, 1, 0]); r_in_line_positive{4,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, -1, 0]/norm([0, -1, 0]); r_in_line_positive{5,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 0, 1]/norm([0, 0, 1]); r_in_line_positive{6,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 0, -1]/norm([0, 0, -1]); elseif q_positive(n_q_positive) == 2 fprintf(1,'* Charge: q = 2 fprintf(1,'* Number of field lines: 8 (Cube) N_line_positive(n_q_positive) = floor(8) r_in_line_positive{1,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_positive{2,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 1, 1]/norm([-1, 1, 1]); r_in_line_positive{3,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); r_in_line_positive{4,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -1, -1]/norm([-1, -1, -1]);

4 r_in_line_positive{5,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, -1, 1]/norm([1, -1, 1]); r_in_line_positive{6,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_positive{7,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 1, -1]/norm([1, 1, -1]); r_in_line_positive{8,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); elseif q_positive(n_q_positive) == 3 fprintf(1,'* Charge: q = 3 fprintf(1,'* Number of field lines: 12 (Icosahedron) N_line_positive(n_q_positive) = floor(12) r_in_line_positive{1,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 1, phi]/norm([0, 1, phi]); r_in_line_positive{2,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, -1, phi]/norm([0, -1, phi]); r_in_line_positive{3,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 1, -phi]/norm([0, 1, -phi]); r_in_line_positive{4,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, -1,-phi]/norm([0, -1,-phi]); r_in_line_positive{5,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, phi, 0]/norm([1, phi, 0]); r_in_line_positive{6,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, -phi, 0]/norm([1, -phi, 0]); r_in_line_positive{7,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, phi, 0]/norm([-1, phi, 0]); r_in_line_positive{8,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -phi, 0]/norm([-1, -phi, 0]); r_in_line_positive{9,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[phi, 0, 1]/norm([phi, 0, 1]); r_in_line_positive{10,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-phi, 0, 1]/norm([-phi, 0, 1]); r_in_line_positive{11,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[phi, 0, -1]/norm([phi, 0, -1]); r_in_line_positive{12,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-phi, 0, -1]/norm([-phi, 0, -1]); elseif q_positive(n_q_positive) == 5 fprintf(1,'* Charge: q = 5 fprintf(1,'* Number of field lines: 20 (Dodecahedron) N_line_positive(n_q_positive) = floor(20) r_in_line_positive{1,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_positive{2,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 1, 1]/norm([-1, 1, 1]); r_in_line_positive{3,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); r_in_line_positive{4,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, -1, -1]/norm([-1, -1, -1]); r_in_line_positive{5,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, -1, 1]/norm([1, -1, 1]); r_in_line_positive{6,n_q_positive} = rq_positive{1,n_q_positive}...

5 + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_positive{7,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1, 1, -1]/norm([1, 1, -1]); r_in_line_positive{8,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); r_in_line_positive{9,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 1/phi, phi]/norm([0, 1/phi, phi]); r_in_line_positive{10,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, -1/phi, phi]/norm([0, -1/phi, phi]); r_in_line_positive{11,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, 1/phi, -phi]/norm([0, 1/phi, -phi]); r_in_line_positive{12,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[0, -1/phi,-phi]/norm([0, -1/phi,-phi]); r_in_line_positive{13,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1/phi, phi, 0]/norm([1/phi, phi, 0]); r_in_line_positive{14,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[1/phi, -phi, 0]/norm([1/phi, -phi, 0]); r_in_line_positive{15,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1/phi, phi, 0]/norm([-1/phi, phi, 0]); r_in_line_positive{16,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-1/phi, -phi, 0]/norm([-1/phi, -phi, 0]); r_in_line_positive{17,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[phi, 0, 1/phi]/norm([phi, 0, 1/phi]); r_in_line_positive{18,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-phi, 0, 1/phi]/norm([-phi, 0, 1/phi]); r_in_line_positive{19,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[phi, 0, -1/phi]/norm([phi, 0, -1/phi]); r_in_line_positive{20,n_q_positive} = rq_positive{1,n_q_positive}... + epsilon*[-phi, 0, -1/phi]/norm([-phi, 0, -1/phi]); tocontinue = input('press Enter to Continue: '); q_positive_net = q_positive_net + q_positive(n_q_positive) %% Define negative charges: fprintf(1,'* Specify the NUMBER of negative charges: N_q_negative = input('enter N_q_negative NUMBER: ') fprintf(1,'* Available charge values: fprintf(1,'* -1, -1.5, -2, -3, -5 % tocontinue = input('press Enter to Continue: '); fprintf(1,'* Begin specifying negative charges and their positions... q_negative = zeros(1,n_q_negative); rq_negative = cell(1,n_q_negative); N_line_negative = zeros(1,n_q_negative) q_negative_net = 0; for n_q_negative = 1:N_q_negative

6 q_negative n_q_negative q_negative(n_q_negative) = input('enter CHARGE q_negative: ') rq_negative{1,n_q_negative} = input('enter rq_negative POSITION VECTOR: ') if q_negative(n_q_negative) == -1 fprintf(1,'* Charge: q = -1 fprintf(1,'* Number of field lines: 4 (Tetrahedron) N_line_negative(n_q_negative) = floor(4) r_in_line_negative{1,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_negative{2,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_negative{3,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); r_in_line_negative{4,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); elseif q_negative(n_q_negative) == -1.5 fprintf(1,'* Charge: q = -1.5 fprintf(1,'* Number of field lines: 6 (Octahedron) N_line_negative(n_q_negative) = floor(6) r_in_line_negative{1,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 0, 0]/norm([1, 0, 0]); r_in_line_negative{2,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 0, 0]/norm([-1, 0, 0]); r_in_line_negative{3,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 1, 0]/norm([0, 1, 0]); r_in_line_negative{4,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, -1, 0]/norm([0, -1, 0]); r_in_line_negative{5,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 0, 1]/norm([0, 0, 1]); r_in_line_negative{6,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 0, -1]/norm([0, 0, -1]); elseif q_negative(n_q_negative) == -2 fprintf(1,'* Charge: q = -2 fprintf(1,'* Number of field lines: 8 (Cube) N_line_negative(n_q_negative) = floor(8) r_in_line_negative{1,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_negative{2,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 1, 1]/norm([-1, 1, 1]); r_in_line_negative{3,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); r_in_line_negative{4,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -1, -1]/norm([-1, -1, -1]);

7 r_in_line_negative{5,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, -1, 1]/norm([1, -1, 1]); r_in_line_negative{6,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_negative{7,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 1, -1]/norm([1, 1, -1]); r_in_line_negative{8,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); elseif q_negative(n_q_negative) == -3 fprintf(1,'* Charge: q = -3 fprintf(1,'* Number of field lines: 12 (Icosahedron) N_line_negative(n_q_negative) = floor(12) r_in_line_negative{1,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 1, phi]/norm([0, 1, phi]); r_in_line_negative{2,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, -1, phi]/norm([0, -1, phi]); r_in_line_negative{3,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 1, -phi]/norm([0, 1, -phi]); r_in_line_negative{4,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, -1,-phi]/norm([0, -1,-phi]); r_in_line_negative{5,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, phi, 0]/norm([1, phi, 0]); r_in_line_negative{6,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, -phi, 0]/norm([1, -phi, 0]); r_in_line_negative{7,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, phi, 0]/norm([-1, phi, 0]); r_in_line_negative{8,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -phi, 0]/norm([-1, -phi, 0]); r_in_line_negative{9,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[phi, 0, 1]/norm([phi, 0, 1]); r_in_line_negative{10,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-phi, 0, 1]/norm([-phi, 0, 1]); r_in_line_negative{11,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[phi, 0, -1]/norm([phi, 0, -1]); r_in_line_negative{12,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-phi, 0, -1]/norm([-phi, 0, -1]); elseif q_negative(n_q_negative) == -5 fprintf(1,'* Charge: q = -5 fprintf(1,'* Number of field lines: 20 (Dodecahedron) N_line_negative(n_q_negative) = floor(20) r_in_line_negative{1,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 1, 1]/norm([1, 1, 1]); r_in_line_negative{2,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 1, 1]/norm([-1, 1, 1]); r_in_line_negative{3,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -1, 1]/norm([-1, -1, 1]); r_in_line_negative{4,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, -1, -1]/norm([-1, -1, -1]); r_in_line_negative{5,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, -1, 1]/norm([1, -1, 1]); r_in_line_negative{6,n_q_negative} = rq_negative{1,n_q_negative}...

8 + epsilon*[1, -1, -1]/norm([1, -1, -1]); r_in_line_negative{7,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1, 1, -1]/norm([1, 1, -1]); r_in_line_negative{8,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1, 1, -1]/norm([-1, 1, -1]); r_in_line_negative{9,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 1/phi, phi]/norm([0, 1/phi, phi]); r_in_line_negative{10,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, -1/phi, phi]/norm([0, -1/phi, phi]); r_in_line_negative{11,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, 1/phi, -phi]/norm([0, 1/phi, -phi]); r_in_line_negative{12,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[0, -1/phi,-phi]/norm([0, -1/phi,-phi]); r_in_line_negative{13,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1/phi, phi, 0]/norm([1/phi, phi, 0]); r_in_line_negative{14,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[1/phi, -phi, 0]/norm([1/phi, -phi, 0]); r_in_line_negative{15,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1/phi, phi, 0]/norm([-1/phi, phi, 0]); r_in_line_negative{16,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-1/phi, -phi, 0]/norm([-1/phi, -phi, 0]); r_in_line_negative{17,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[phi, 0, 1/phi]/norm([phi, 0, 1/phi]); r_in_line_negative{18,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-phi, 0, 1/phi]/norm([-phi, 0, 1/phi]); r_in_line_negative{19,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[phi, 0, -1/phi]/norm([phi, 0, -1/phi]); r_in_line_negative{20,n_q_negative} = rq_negative{1,n_q_negative}... + epsilon*[-phi, 0, -1/phi]/norm([-phi, 0, -1/phi]); tocontinue = input('press Enter to Continue: '); q_negative_net = q_negative_net + q_negative(n_q_negative) %% Net charge: q_net = q_positive_net + q_negative_net %% Plotting all charges: fprintf(1,'* Plotting all the charges... fprintf(1,'* Calculating and plotting the field lines... tocontinue = input('press Enter to Continue: '); for n_q_positive = 1:N_q_positive plot3(rq_positive{1,n_q_positive}(1),... rq_positive{1,n_q_positive}(2),... rq_positive{1,n_q_positive}(3),... 'r*',... 'linewidth',1.0) hold on

9 for n_q_negative = 1:N_q_negative plot3(rq_negative{1,n_q_negative}(1),... rq_negative{1,n_q_negative}(2),... rq_negative{1,n_q_negative}(3),... 'b*',... 'linewidth',1.0) hold on %% Check which lines to plot: positive or negative: if q_net >= 0 %% Build all lines for each of the positive charges: for n_q_positive = 1:N_q_positive for n_line_positive = 1:N_line_positive(n_q_positive) r_in = r_in_line_positive{n_line_positive,n_q_positive}; Calculating a given field line: N_step; %%% Chosen above x = zeros(1,n_step); y = zeros(1,n_step); z = zeros(1,n_step); Ex = zeros(1,n_step); Ey = zeros(1,n_step); Ez = zeros(1,n_step); Ex_quivered = zeros(1,n_step); Ey_quivered = zeros(1,n_step); Ez_quivered = zeros(1,n_step); dr = [0, 0, 0]; %%% vector r = r_in; for n_step = 1:N_step n_step; New position: x(n_step) = r(1) + (n_step-1)*dr(1); y(n_step) = r(2) + (n_step-1)*dr(2); z(n_step) = r(3) + (n_step-1)*dr(3); r = [x(n_step), y(n_step), z(n_step)]; %%% vector E_net = [0, 0, 0]; for m_q_positive = 1:N_q_positive r_from_q = r - rq_positive{1,m_q_positive}; %%% vector r_from_q_norm = norm(r_from_q); r_from_q_hat = r_from_q/r_from_q_norm; %%% vector E_from_q =... q_positive(m_q_positive)*r_from_q_hat/r_from_q_norm^2; Net E-field: E_net = E_net + E_from_q;

10 if N_q_negative > 0 for m_q_negative = 1:N_q_negative r_from_q = r - rq_negative{1,m_q_negative}; %%% vector r_from_q_norm = norm(r_from_q); r_from_q_hat = r_from_q/r_from_q_norm; %%% vector E_from_q =... q_negative(m_q_negative)*r_from_q_hat/r_from_q_norm^2; Net E-field: E_net = E_net + E_from_q; Ex(n_step) = E_net(1); Ey(n_step) = E_net(2); Ez(n_step) = E_net(3); E_net_norm = norm(e_net); E_net_normalized = E_net/E_net_norm; Ex_quivered(n_step) = E_net_normalized(1); Ey_quivered(n_step) = E_net_normalized(2); Ez_quivered(n_step) = E_net_normalized(3); Displacement along the E-line: step_factor; %%% Chosen above Sign of q determines E-line propagation direction: step_r_factor =... step_factor*q_positive(n_q_positive)/abs(q_positive(n_q_positive)); dr = step_r_factor*e_net_normalized; %%% vector quiver3(x,y,z,ex_quivered,ey_quivered,ez_quivered,0.25,'r-',... 'linewidth',1.0) hold on elseif q_net < 0 %% Build all lines for each of the negative charges: for n_q_negative = 1:N_q_negative for n_line_negative = 1:N_line_negative(n_q_negative) r_in = r_in_line_negative{n_line_negative,n_q_negative}; Calculating a given field line: N_step; %%% Chosen above x = zeros(1,n_step); y = zeros(1,n_step); z = zeros(1,n_step); Ex = zeros(1,n_step); Ey = zeros(1,n_step);

11 Ez = zeros(1,n_step); Ex_quivered = zeros(1,n_step); Ey_quivered = zeros(1,n_step); Ez_quivered = zeros(1,n_step); dr = [0, 0, 0]; %%% vector r = r_in; for n_step = 1:N_step n_step; New position: x(n_step) = r(1) + (n_step-1)*dr(1); y(n_step) = r(2) + (n_step-1)*dr(2); z(n_step) = r(3) + (n_step-1)*dr(3); r = [x(n_step), y(n_step), z(n_step)]; %%% vector E_net = [0, 0, 0]; for m_q_negative = 1:N_q_negative r_from_q = r - rq_negative{1,m_q_negative}; %%% vector r_from_q_norm = norm(r_from_q); r_from_q_hat = r_from_q/r_from_q_norm; %%% vector E_from_q =... q_negative(m_q_negative)*r_from_q_hat/r_from_q_norm^2; Net E-field: E_net = E_net + E_from_q; if N_q_positive > 0 for m_q_positive = 1:N_q_positive r_from_q = r - rq_positive{1,m_q_positive}; %%% vector r_from_q_norm = norm(r_from_q); r_from_q_hat = r_from_q/r_from_q_norm; %%% vector E_from_q =... q_positive(m_q_positive)*r_from_q_hat/r_from_q_norm^2; Net E-field: E_net = E_net + E_from_q; Ex(n_step) = E_net(1); Ey(n_step) = E_net(2); Ez(n_step) = E_net(3); E_net_norm = norm(e_net); E_net_normalized = E_net/E_net_norm; Ex_quivered(n_step) = E_net_normalized(1); Ey_quivered(n_step) = E_net_normalized(2); Ez_quivered(n_step) = E_net_normalized(3); Displacement along the E-line: step_factor; %%% Chosen above Sign of q determines E-line propagation direction: step_r_factor =... step_factor*q_negative(n_q_negative)/abs(q_negative(n_q_negative)); dr = step_r_factor*e_net_normalized; %%% vector

12 quiver3(x,y,z,ex_quivered,ey_quivered,ez_quivered,0.25,'b-',... 'linewidth',1.0) hold on grid on axis tight set(gca,'box','on'); %%% BOX set(gca,'fontsize',12) %%% the fontsize of the axes tick labels xlabel('x'), ylabel('y'), zlabel('z') title({... ['q_{net} = ',num2str(q_net),... ' N_{+} = ',num2str(n_q_positive),... ' q_{+} = \{',num2str(q_positive),'\}'... ' N_{-} = ',num2str(n_q_negative),... ' q_{-} = \{',num2str(q_negative),'\}'... ],... ['\epsilon = ', num2str(epsilon),... ' N_{step} = ',num2str(n_step),... ' step_{factor} = ',num2str(step_factor),... ],... }) hold off fprintf(1,'* Displaying all charges and their positions: q_net q_positive, celldisp(rq_positive) q_negative, celldisp(rq_negative) fprintf(1,'* Calculation and plotting the field lines is complete. fprintf(1,'* fprintf(1,'* END OF CODE.