Motion of charger particles perpendicular to magnetic field | part 1: cyclotron frequency

 

The cyclotron frequency is a term used in electromagnetism to describe the frequency at which a charged particle orbits in a magnetic field. It is calculated using the equation:

ω = qB/m

where ω is the cyclotron frequency, q is the charge of the particle, B is the strength of the magnetic field, and m is the mass of the particle. 

We have F= q v*B

Angle between v and B is right angle so 

F=qvB

Now the path of charged particle is circular,

The magnetic force is this balanced with centripetal force. Which is to say magbetic acceleration balances centripetal acceleration.

This gives R=mv/qB where R is the radius of electron path.

We rearrange the equation to find 

v=RqB/m

T=2pir/v

On putting values,

T=2pi m/qB

Since frequency is equal to reciprocal of time so, f=qB/2pi m

Angular frequency is related to orbital frequency as,

ω=2 pi f

So on putting value of f we get

ω= qB/m which is proved as required. 

Applications of cyclotron frequency: 

This frequency depends only on the charge and mass of the particle, as well as the strength of the magnetic field, but not on the particle's initial velocity or position. The cyclotron frequency is commonly used in the study of particle acceleration, particularly in particle accelerators such as cyclotrons and synchrotrons. It helps determine the path and speed at which charged particles move in a magnetic field, allowing for precise control of their motion for experimental purposes.