1. Charge in a constant uniform electric & magnetic field
Section 22

2. We consider non-relativistic motion only. (Why
We consider non-relativistic motion only. (Why? Because otherwise it’s too hard? Or is there a reason that this is all that matters?)
Then, p = mv
Choose Z || H
Choose X &Y so that H & E lie in the YZ plane Z H E Y X

3. Equation of motion
Non-relativistic case

4. The Z equation is the easiest.
Integrate twice
This is the equation for uniform acceleration in the z direction.

5. Add X and Y equations

6. Cyclotron frequency in Gaussian units
Sum of the solutions of the homogeneous equation and the particular solution.

7. Homogeneous equation
Particular solution

8. Complete solution
Complex constant
Real
Now shift the origin of time by a/w to eliminate constant phase.

9. Same as if a = 0. Then b = a.
At t = 0,
Velocity components are periodic functions of time.

10. Time-averaged values
Average X velocity is constant: the drift velocity
Perpendicular to both E & H X Y Z H E vD

11. Drift velocity H

12. All derivations in this section assume v << c.
But both fields can be large.
In SI units Ey << c B = c m0 H.

13. Trajectory The velocity equations were Integrate
The constant is determined by the initial conditions.

14. What is a cycloid?
What is a trochoid?

17. Trajectory (ignoring constant acceleration along z due to Ez)X Y Z B E

18. Which trochoid is a cycloid?1 2 3