1. Physics 319 Classical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 4
G. A. Krafft
Jefferson Lab

2. Motion in a Uniform Magnetic Field
Magnetic (Lorentz) force also depends on velocity
Solve this equation for a uniform (independent of position) field

3. Equations of Motion Newton’s Law
z component gives uniform velocity in z direction
Cyclotron (angular) frequency
Conserved transverse energy

4. Easy Solution with Complex Numbers
Define η = vx +ivy
A is a complex number giving initial conditions. Do these formal manipulations make sense?
Complex plane
imaginary axis
real axis

5. Complex Exponentials Usual power series works for complex numbers too
Euler’s formula
Consistent with the polar representation!

6. Complex Representation
For z pure imaginary, has unit length
Integrate to get positions
Radius of transverse orbit
Constants of motion (4) are X, Y, a, and δ. δ is the phase of the oscillation

7. Picture of Motion
imaginary axis
real axis