PHYSICS 133 General Physics: Electromagnetism

PHYSICS 202 – Physics and the Computer

Projects

Point particle near a fixed dipole

In particle accelerators and particle detectors we often control the path of particles using charged conductors. Let’s consider two conducting spheres radius = 1.0 mm separated by a distance, d = 2 cm, and charged to +/- Q = +/- 1 nC. Let the positive charge reside at +d/2 and the negative charge reside at –d/2 along the x-axis.

Write reusable functions to calculate the potential, V, and the electric fields, Ex, and Ey for the conductors. The new functions may use the dipolePotential.m and dipoleField.m functions and then set the potential to a constant inside the conductor, and the electric field to NaN inside the conductor so that particles do not propagate further after hitting the surface of the conductor. Plot the results using:

Q = 1e-9;

d = 0.01;

r = 0.001;

[x,y] = meshgrid(-.02:0.0003:0.02, -.02:0.0003:0.02);

V = dipoleConductorPotential(x,y,Q,d,r);

[Ex, Ey] = dipoleConductorField(x,y,Q,d,r);

figure(1)

surf(V) % plot the potential surface

figure(2)

levels = [8 4 2 1 0.5 0.25 0 -0.25 -0.5 -1 -2 -4 -8]*1e3;

contour(x,y,V,levels); % contour plot of the potential surface

hold on

quiver(x,y,Ex,Ey,2); % overlay E-field vectors

hold off;

Assume an electron (mass m=9.109e-31 kg, charge q=-1.602e-19 C) is initially located in the plane at (x0,y0) and released with an initial velocity (vx0, vy0).

a) Write a routine to determine the position as a function of time for the electron. You should be solving for x(t), y(t), vx(t), and vy(t).

b) Try the following test cases to debug the program.

• (x0,y0)=(0 cm, 0 cm) and (vx0,vy0)=(0 m/s, 0 m/s)
• (x0,y0)=(0 cm, 2 cm) and (vx0,vy0)=(0 m/s, 0 m/s)
• (x0,y0)=( -0.5 cm, 2 cm) and (vx0,vy0)=(0 m/s, 0 m/s)

Plot x(t) and y(t) vs. t on the same axes, vx(t) and vy(t) vs. t on the same axes to help debug the program. When things start working, plot the trajectory, y vs. x.

c) When everything looks good, add the trajectory to the graph (figure 2 above) of electric potential contours and electric field vectors.

d) Use quiver(x,y,vx,vy) to add the velocity vectors to the plot.

e) Take the derivative of vx and vy to find ax and ay. Add these to the plot using Matlab’s quiver function. Explain with words and the graphics you’ve created why the electric fields and acceleration vectors are aligned. Are there places where this is not true along your trajectory? (quiver won’t plot NaN. You’ll need to find(~isnan(ax) & ~isnan(ay)) and plot just these quivers. Also, you’ll need to modify finiteDifference.m so that if dx or dy is NaN, then it returns NaN.)

f) How does the energy vary with time? Plot E vs t. Explain the relationship between the contours and the velocities.

g) Choose several different starting locations and initial velocities. Be sure to include some initial locations and velocities where you know the answer so you can check your results.

h) If you have time, plot the trajectory of a proton. You’ll need to adjust some parameters due to the mass of the proton. These differences in trajectories and timing are exploited in particle detectors to distinguish between protons and electrons.

mjm 5/19/19 P-DipoleForce