Screen shot of yellow test charge moving through green ring of charge.

# Charge Ring Trajectory Model

The EJS Charge Ring Trajectory Model shows the trajectory of a test charge through a ring of charges. The size of the ring and the initial conditions of the test charge can be changed. Users can examine and edit the model if Ejs is installed. Information about EJS is available at: <http://www.um.es/fem/Ejs/>.

## Questions:

1. Run the simulation. You can click-drag in the the window to rotate the view and shift-click to zoom in and out. The plot shows the x-component of position (blue) and velocity (red) of the yellow test charge as a function of time. The convention is that a test charge is positive, but of small enough size so that its own electric field can be neglected. Therefore, the simulation only shows the electric field due to the green charges. Are the green charges positive or negative? Explain.
2. Change the radius of the ring of green charges and observe the motion. Why does the test charge stay on the x-axis?
3. If you Pause the simulation and move the charge slightly off-axis, what happens? Why?
4. Reset the simulation and run it with the initial parameters (test charge on the x-axis and a ring radius of 8 cm). Look at the data plot. What is the period of oscillation?
5. Explain why, for larger values of the ring radius and motion near the ring, the motion is sinusoidal (it looks like simple harmonic motion). Hint: Start with the equation of the electric field on axis of a ring of charge and compare it to the the force for simple harmonic motion. Show that the frequency of oscillation (of this simple harmonic oscillator) is equal to ω=(qQ/4πε0mR3)1/2 where q is the charge of the test charge, Q, the magnitude of the total charge on the ring and R, the radius of the ring. Therefore, if the total charge on the ring is one million times the charge of the test charge and the test charge has a mass of 1 μg (=10-9 kg), what is the magnitude of the green and yellow charges?

### References:

- Haliday, Resnick and Walker, Fundamentals of Physics, 8th edition, Chapter 22 (2007).

### Credits:

The Charge Ring Trajectory Model was created by Anne J Cox using the Easy Java Simulations (EJS) authoring and modeling tool.

You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file. Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS. You must, of course, have EJS installed on your computer.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.