Classical Dynamics of Particles and Systems · Problem 2-22

Oscillations — Marion & Thornton

Physics — Mechanics · lorentz-force-charged-particle

Problem

The motion of a charged particle in an electromagnetic field can be obtained from the Lorentz equation* for the force on a particle in such a field. If the electric field vector is

\mathbf{E}

and the magnetic field vector is

\mathbf{B}

, the force on a particle of mass

m

that carries a charge

q

and has a velocity

\mathbf{v}

is given by

\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}

where we assume that

v \ll c

(speed of light). (a) If there is no electric field and if the particle enters the magnetic field in a direction perpendicular to the lines of magnetic flux, show that the trajectory is a circle with radius

r = \frac{mv}{qB} = \frac{v}{\omega_c}

where

\omega_c \equiv qB/m

is the cyclotron frequency. (b) Choose the

z

-axis to lie in the direction of

\mathbf{B}

and let the plane containing

\mathbf{E}

and

\mathbf{B}

be the

yz

-plane. Thus

\mathbf{B} = B\mathbf{k}, \quad \mathbf{E} = E_y\mathbf{j} + E_z\mathbf{k}

Show that the

z

component of the motion is given by

z(t) = z_0 + \dot{z}_0 t + \frac{qE_z}{2m}t^2

where

z(0) \equiv z_0

and

\dot{z}(0) \equiv \dot{z}_0

. (c) Continue the calculation and obtain expressions for

\dot{x}(t)

and

\dot{y}(t)

. Show that the time averages of these velocity components are

\langle \dot{x} \rangle = \frac{E_y}{B}, \quad \langle \dot{y} \rangle = 0

(Show that the motion is periodic and then average over one complete period.) (d) Integrate the velocity equations found in (c) and show (with the initial conditions

x(0) = -A/\omega_c

\dot{x}(0) = E_y/B

y(0) = 0

\dot{y}(0) = A

) that

x(t) = \frac{-A}{\omega_c}\cos\omega_c t + \frac{E_y}{B}t, \quad y(t) = \frac{A}{\omega_c}\sin\omega_c t

These are the parametric equations of a trochoid. Sketch the projection of the trajectory on the

xy

-plane for the cases (i)

A > |E_y/B|

, (ii)

A < |E_y/B|

, and (iii)

A = |E_y/B|

. (The last case yields a cycloid.)

Related theory

Lorentz Force Particle Acceleration

Browse the problem library