Classical Mechanics (Kibble & Berkshire) · Problem 4.33

Linear oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · circular-motion-magnetron

Problem

The magnetron An electron of mass

m

and charge

-e

is moving under the combined influence of a uniform electric field

E_0\boldsymbol{j}

and a uniform magnetic field

B_0\boldsymbol{k}

. Initially the electron is at the origin and is moving with velocity

u\boldsymbol{i}

. Show that the trajectory of the electron is given by

x = a(\Omega t) + b\sin\Omega t, \quad y = b(1-\cos\Omega t), \quad z = 0,

where

\Omega = eB_0/m

a = E_0/\Omega B_0

and

b = (uB_0 - E_0)/\Omega B_0

. Use computer assistance to plot typical paths of the electron for the cases

a < b

a = b

and

a > b

. [The general path is called a trochoid, which becomes a cycloid in the special case

a = b

. Cycloidal motion of electrons is used in the magnetron vacuum tube, which generates the microwaves in a microwave oven.]