## Mastering Physics Solutions: Charge Moving in a Cyclotron Orbit

Charge Moving in a Cyclotron Orbit

Part A = -y direction
Part B = remain perpendicular to the direction of motion
Part C = remain constant
Part D = ω = qB0/m

Solutions Below:

Learning Goal: To understand why charged particles move in circles perpendicular to a magnetic field and why the frequency is an invariant.
A particle of charge q and mass m moves in a region of space where there is a uniform magnetic field B = Bẑ (i.e., a magnetic field of magnitude B in the +z direction). In this problem, neglect any forces on the particle other than the magnetic force.

Part A

At a given moment the particle is moving in the +x direction (and the magnetic field is always in the +z direction). If q is positive, what is the direction of the force on the particle due to the magnetic field?

Use the right hand rule.

-y direction

Part B

This force will cause the path of the particle to curve. Therefore, at a later time, the direction of the force will:

• have a component along the direction of motion
• remain perpendicular to the direction of motion
• have a component against the direction of motion
• first have a component along the direction of motion; then against it; then along it; etc.

remain perpendicular to the direction of motion

Part C

The fact that the magnetic field generates a force perpendicular to the instantaneous velocity of the particle has implications for the work that the field does on the particle. As a consequence, if only the magnetic field acts on the particle, its kinetic energy will:

• increase over time
• decrease over time
• remain constant
• oscillate

remain constant

Part D

If the resulting trajectory of the charged particle is a circle, what is omega, the angular frequency of the circular motion?

Express ω in terms of q, m, and B.

This one is kind of tricky.

Fmag = qvB

Centripetal acceleration is:

ar = v2/R

and centripetal force:

Fc = mv2/R

And velocity is:

v = ωR

Since the magnetic force is the centripetal force:

Fmag = Fc
qvB = mv2/R
qB = mv/R

Substitute for velocity:

qB = mωR/R
qB = mω

ω = qB/m

Since the problem wants things in terms of B0:

ω = qB0/m
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