# E x B Field Trajectory

The EJS E x B Field Trajectory Model shows the motion of a charged particle in a constant electric and magnetic field. The model is initially incomplete and students must complete the model (by adding the appropriate equations of motion). Ejs must be installed to complete the model. Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.

## Exercises:

1. Run the simulation. Notice that changing the magnetic field does not impact the motion. That is because the Lorentz force law is not complete. Right-click on the simulation and Open EJS Model. Choose Model and look at the Evolution page. Right now, the force law governing the motion is simply:

F = q E or a = (q/m)*E

The evolution page needs six equations for this: three to define velocity and three to define acceleration. Why are there three of each?

2. In order for the magnetic field to impact the motion, you need to include the magnetic field in the Lorentz equation:

F = q (E + v x B)

the x-component of this equation is

Fx = q*(Ex + vy*Bz - vz*By).

Why? What are the rest of the components?

Fy = Fz =

3. Complete the EJS model using the appropriate acceleration equations.
4. To get the words, Incomplete Model off the screen, go to the Variables page, choose the Display tab and change ModelComplete from false to true.
5. Test to see that you have set up the model correctly (for q/m = 1):

- If E = 0, Bx = By = 0 and Bz = 1 (or B = 1k) and initially vx = 1, vy = 0 and vz = 0 (v = 1i), you should see a circular trajectory. Why? What are some other configurations that should give a circular trajectory? Test them and verify that they are circular. How can you generate a circle of a smaller radius? Explain.
- If E = 1i, B = 1k and initially v = -1j, what is the trajectory? Why?
- If E = 1i, B = 1i, and initially v =0, the motion is the same no matter the value of Bx. Why?
- If E = 1i, B = 1k and intially v = 0, why won't the particle change its z-component of motion? Check that this is the case for your simulation.
- What configuration matches (qualitatively) the cathrode tube demo from class?

### References:

- Giancoli, Physics for Scientists and Engineers, 4th edition, Chapter 27 (2008).
- D.J. Griffiths, Introduction to Electrodynamics, 3rd edition, pp. 204-207 (1999).

### Credits:

The E x B Trajectory Model and Exercises were 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/>.