MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Cyclotron Motion Solutions:
A charged particle with positive charge q is moving with speed v in a uniform magnetic
!
field B pointing into the figure below.
(1) Draw a sketch of the motion of the particle showing the forces acting on the
particle at various positions in its orbit.
(2) Find the radius R of the orbit.
(3) Find the period T that it takes the particle first return to it’s position shown in the
figure above.
(4) Find the cyclotron angular frequency ! of the motion.
!
At the instant shown in the figure above the particle is moving with velocity v = v ĵ . The
!
uniform magnetic field points into the page, so B = ! B k̂ with B > 0 . The force on the
particle at that instant is
!
! !
F = qv ! B = q(v ĵ) ! (" B k̂) = "qvB î .
Because this force is perpendicular in to the velocity, it only changes the direction of the
velocity and not the magnitude hence the particle moves in a circular orbit as shown in
the figure below.
(2) Find the radius R of the orbit.
Answer: The orbit is circular, so Newton’s second Law becomes qvB = mv 2 / R . Thus
the radius of the orbit is
mv
R=
.
qB
(3) Find the period T that it takes the particle first return to it’s position shown in the
figure above.
Answer: The time T it takes the particle to complete a semicircular path from S to P is
T=
2! R 2! m
=
.
v
qB
(4) Find the cyclotron angular frequency ! of the motion.
Answer:
!=
2" v vqB qB
= =
=
T
R mv
m