Question 9: What is cyclotron frequency of a charged particle q, of mass m moving in a magnetic field B?

## Cyclotron Frequency

Cyclotron frequency is the frequency of a charged particle moving perpendicular to the direction of the uniform magnetic field.

#### Mathematical expression for cyclotron frequency

When a charged particle q is released in a constant magnetic field B with a velocity v, it experiences a force given by the equation

Direction of the force is normal to the plane of v and B.

Suppose a particle of charge q is shot with velocity v into a uniform magnetic field B in a direction perpendicular to B. Let the mass of the particle is m. The magnetic force acting on the particle is

The direction of the force will be perpendicular to both v and B. This constant and normal in direction force will make the particle move in a circular path of radius r. The magnetic force provides the necessary centripetal force (mv2/r). Hence,

Here ω is the angular velocity of the charged particle.

We know that ω = 2πf, therefore, equating both equations

‘f’ is called the cyclotron frequency of the particle. The above equation gives the cyclotron frequency of a charged particle q having mass m and moving in a uniform magnetic field B.