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 = qB_{0}/m

**Solutions Below:**

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

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

- increase over time
- decrease over time
- remain constant
- oscillate

remain constant

Part D

Express ω in terms of q, m, and B.

This one is kind of tricky.

F_{mag} = qvB

Centripetal acceleration is:

a_{r} = v^{2}/R

and centripetal force:

F_{c} = mv^{2}/R

And velocity is:

v = ωR

Since the magnetic force is the centripetal force:

F_{mag} = F_{c}

qvB = mv^{2}/R

qB = mv/R

Substitute for velocity:

qB = mωR/R

qB = mω

ω = qB/m

Since the problem wants things in terms of B_{0}:

ω = qB_{0}/m