Introduction:
The cyclotron frequency is a fundamental concept in physics that describes the rotation of charged particles in a magnetic field. It is an essential parameter in the study of particle accelerators and magnetic resonance imaging (MRI) technology. This frequency can be derived by considering the motion of a charged particle in a uniform magnetic field.

Motion of a Charged Particle:
When a charged particle with charge q and mass m is placed in a uniform magnetic field B, it experiences a magnetic force given by the equation F = qvB, where v is the velocity of the particle. This force acts as a centripetal force that keeps the particle moving in a circular path.

Centripetal Force:
The centripetal force required to keep the particle in a circular path can be given by the equation F = (mv^2)/r, where r is the radius of the circular path. Equating these two forces, we have qvB = (mv^2)/r.

Deriving the Cyclotron Frequency:
By rearranging the equation, we get qB = (mv)/r. Here, we can express the velocity v in terms of the radius r and the angular velocity ω.

Angular Velocity:
The angular velocity ω is defined as the rate at which the particle rotates in radians per unit time. It can be calculated using the equation ω = v/r.

Substituting for v:
Substituting the value of v in terms of ω and r in the previous equation, we get qB = mωr.

Cyclotron Frequency:
The cyclotron frequency, denoted by ωc, is defined as the angular frequency of the circular motion of the charged particle. It is given by the equation ωc = qB/m.

Conclusion:
The cyclotron frequency is derived by considering the motion of a charged particle in a uniform magnetic field. It represents the angular frequency of the circular motion and is given by the equation ωc = qB/m. Understanding the cyclotron frequency is crucial in various fields of physics, such as particle accelerators and MRI technology.