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

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

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

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

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

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

Picture of Motion imaginary axis real axis