Estimate the total path length traveled by a deuteron in a...

Estimate the total path length traveled by a deuteron in a cyclotron of radius $53 \mathrm{~cm}$ and operating frequency $12 \mathrm{MHz}$ during the (entire) acceleration process. Assume that the accelerating potential between the dees is $80 \mathrm{kV}$.

Key Concepts

Cyclotron Operation

A cyclotron is a type of particle accelerator that uses a constant magnetic field and an oscillating electric field to accelerate charged particles along a spiral trajectory. The magnetic field forces the charged particle to move in a circular path, while the alternating electric field provides energy boosts each time the particle crosses a gap between the dees (electrode structures).

Charged Particle Kinematics in Magnetic Fields

When a charged particle moves in a perpendicular magnetic field, it experiences the Lorentz force which causes it to follow a circular path with a radius that is dependent on its momentum and the magnetic field strength. This concept is fundamental to understanding the motion of particles in cyclotrons and other magnetic confinement devices.

Energy Gain Per Turn in Particle Accelerators

In a cyclotron, the particle gains energy in discrete amounts as it crosses the gap between the dees. The accelerating potential applied between the dees determines the energy increment per revolution. The cumulative energy gain over many turns dictates how far from the center the particle can spiral before it reaches the desired energy.

Cyclotron Frequency and Resonance

The cyclotron frequency, defined by the ratio qB/(2?m) for a particle of charge q and mass m in a magnetic field B, is the frequency at which the particle orbits in the cyclotron. This frequency is independent of the particle's energy, allowing the accelerator to maintain resonance between the particle’s motion and the applied oscillating electric field for continuous acceleration.

Total Path Length Calculation

The total path length a particle travels in a cyclotron is determined by the sum of the lengths of all the circular segments it traverses. As the particle gains energy with each turn, its orbital radius increases gradually. The overall path length can be estimated by considering the number of revolutions and the circumference corresponding to the evolving radius up to the maximum radius of the cyclotron.

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