The relationship between the time period (T) and the maximum speed (v) of a charged particle in a cyclotron can be expressed through the following considerations:

In a cyclotron, charged particles move in a circular path due to a magnetic field. The magnetic force provides the centripetal force necessary for this circular motion. The radius (r) of the circular path increases as the particle gains speed.

The time period (T) of the circular motion is the time it takes for the particle to complete one full revolution.

The formula for the time period (T) of a charged particle in a cyclotron is:

• where:
• m is the mass of the charged particle
• q is the charge of the particle
• B is the magnetic field strength

The time period is independent of the particle's speed and radius.

The maximum speed (v) of the particle is related to the maximum radius (r) of the cyclotron:

• where:
• r is the radius of the cyclotron Dees

From here, the relationship can be rearranged to show:

In summary, while T is independent of v in the cyclotron's operational principle, the maximum achievable v is constrained by the cyclotron's physical dimensions and the applied magnetic field.

Source: MIT Physics - Cyclotron