Elektrostatik - Aufgabe 2

Gegebene Größen: Ladungen & Punkt

clear all;

q(1).c = 25e-9; %Coulomb

q(1).pos = 0.01*[0 4]; %cm

q(2).c = 0.1e-6;

q(2).pos = 0.01*[0 0];

q(3).c = -50e-9;

q(3).pos = 0.01*[-4 3];

e_0 = 8.854e-12; %C^2/(N m^2)

P = 0.01*[0 3];

windowMin = min([q(:).pos P]);

windowMax = max([q(:).pos P]);

% Plot

pos = vertcat(q(:).pos)';

plot(pos(1,:),pos(2,:),'LineStyle','None','Marker','o','MarkerSize',10,'MarkerFaceColor','b');

box off;

grid on;

hold on;

plot(P(1),P(2),'LineStyle','None','Marker','o','MarkerSize',10,'MarkerFaceColor','k');

hold off;

xlim([windowMin windowMax]);

ylim([windowMin windowMax]);

Teil a)

Kräfte

r_12 = norm(q(1).pos - q(2).pos)
r_12 = 0.0400
F_12 = q(1).c*q(2).c / (4*pi*e_0*r_12^2)
F_12 = 0.0140
r_23 = norm(q(3).pos - q(2).pos)
r_23 = 0.0500
F_23 = q(3).c*q(2).c / (4*pi*e_0*r_23^2)
F_23 = -0.0180
winkel_12 = rad2deg(cart2pol(q(1).pos(1) - q(2).pos(1),q(1).pos(2) - q(2).pos(2)))
winkel_12 = 90
winkel_23 = rad2deg(cart2pol(q(3).pos(1) - q(2).pos(1),q(3).pos(2) - q(2).pos(2)))
winkel_23 = 143.1301
%Winkel aus aufgabe
alpha_23 = 180 - winkel_23;
alpha_12 = 180 - winkel_12;
%Negativ auf der y-Achse!
F_23_buch = [F_23 * cosd(alpha_23) -F_23 * sind(alpha_23)]
F_23_buch = 1×2
   -0.0144    0.0108
F_12_buch = [F_12 * cosd(alpha_12) -F_12 * sind(alpha_12)]
F_12_buch = 1×2
         0   -0.0140
%Da der koordinatenursprung in der ausgesetzten ladung lag, vorzeichen umdrehen!
F_12 = -1 * [F_12 * cosd(winkel_12) F_12 * sind(winkel_12)]
F_12 = 1×2
         0   -0.0140
F_23 = -1 * [F_23 * cosd(winkel_23) F_23 * sind(winkel_23)]
F_23 = 1×2
   -0.0144    0.0108
F_res = F_12 + F_23
F_res = 1×2
   -0.0144   -0.0033

Teil b)

Berechne die Abstände

[X,Y] = meshgrid(linspace(windowMin,windowMax,20)); 
for i = 1:numel(q)
    %Polarkoordinaten (allgemein)
    [q(i).winkel, q(i).rAll] = cart2pol(X-q(i).pos(1),Y - q(i).pos(2));
    
    %Aufgabenspezifisch
    [q(i).winkel_p, q(i).r_p] = cart2pol(P(1) - q(i).pos(1), P(2) - q(i).pos(2));
    q(i).dist = P - q(i).pos;
    %q(i).r = norm(q(i).dist);
    fprintf('rp_%d = %f\n',i,q(i).r_p); 
end
rp_1 = 0.010000
rp_2 = 0.030000
rp_3 = 0.040000
E_feld = @(q,r) q ./ (4*pi*e_0*r.^2);

Beiträge der E-Felder

%E-Felder Beträge
for i = 1:numel(q)
    q(i).E = E_feld(q(i).c,q(i).r_p);
    fprintf('E_%d = %g\n',i, q(i).E);
    
    q(i).EAll = E_feld(q(i).c,q(i).r_p);
end
E_1 = 2.24694e+06
E_2 = 998638
E_3 = -280867

Feldvektoren

E_feldvec_x = @(E,winkel) E .* cos(winkel);
E_feldvec_y = @(E,winkel) E .* sin(winkel);
ExAll = zeros(size(X));
EyAll = zeros(size(Y));
%Feldkomponenten
for i = 1:numel(q)
    q(i).E_vec(1) = E_feldvec_x(q(i).E,q(i).winkel_p);
    q(i).E_vec(2) = E_feldvec_y(q(i).E,q(i).winkel_p);
    disp(q(i).E_vec)
    
    q(i).Ex_All = E_feldvec_x(q(i).EAll,q(i).winkel);
    q(i).Ey_All = E_feldvec_y(q(i).EAll,q(i).winkel);
    
    
    ExAll = ExAll + q(i).Ex_All;
    EyAll = EyAll + q(i).Ey_All;
    
    %quiver(X,Y,q(i).Ex_All,q(i).Ey_All);
end
   1.0e+06 *

    0.0000   -2.2469

   1.0e+05 *

    0.0000    9.9864

   1.0e+05 *

   -2.8087         0

Superposition und Endergebnis

EP_All = vertcat(q(:).E_vec);

EP_Sup = sum(EP_All,1)

EP_Sup = 1×2

10⁶ ×

   -0.2809   -1.2483

EP = norm(EP_Sup)

EP = 1.2795e+06

EAll = sqrt(ExAll.^2 + EyAll.^2);

contourf(X,Y,EAll); hold on;

colormap copper; colorbar;