An early cyclotron at Cornell University was used from the 1930 s to the 1950 s to accelerate

step 1 All right, so we've got a moving proton in a magnetic field, and we would like to know what the

An early cyclotron at Cornell University was used from the...

An early cyclotron at Cornell University was used from the $1930 \mathrm{~s}$ to the $1950 \mathrm{~s}$ to accelerate protons, which would then bombard various nuclei. The cyclotron used a large electromagnet with an iron yoke to produce a uniform magnetic field of $1.3 \mathrm{~T}$ over a region in the shape of a flat cylinder. Two hollow copper dees of inside radius $16 \mathrm{~cm}$ were located in a vacuum chamber in this region. (a) What is the frequency of oscillation necessary for the alternating voltage difference between the dees? (b) What is the kinetic energy of a proton by the time it reaches the outside of the dees? (c) What would be the equivalent voltage necessary to accelerate protons to this energy from rest in one step (say between parallel plates)? (d) If the potential difference between the dees has a magnitude of $10.0 \mathrm{kV}$ each time the protons cross the gap, what is the minimum number of revolutions each proton has to make in the cyclotron?

Key Concepts

Cyclotron

A cyclotron is a type of particle accelerator that uses a constant magnetic field and a rapidly alternating electric field to accelerate charged particles along a spiral path. The particles are injected at the center and experience increasing acceleration with each pass through the gap between the dees, causing them to spiral outward until they reach the desired energy.

Uniform Magnetic Field

A uniform magnetic field is critical in a cyclotron as it provides a consistent centripetal force on the moving charged particles. This consistency ensures that the particles follow a predictable circular path. The magnetic field's strength directly influences the orbital radius and the cyclotron frequency, which is the rate at which the particles complete turns.

Cyclotron Frequency

The cyclotron frequency is the frequency at which a charged particle orbits in a uniform magnetic field, determined by the particle's charge and mass as well as the magnetic field strength. This frequency is fundamental for matching the applied alternating electric field, which must be synchronized with the particle's motion for effective acceleration.

Particle Kinetic Energy

In particle acceleration, the kinetic energy of a charged particle is built up gradually as it gains speed with each pass through the accelerating gap. The concept of kinetic energy links the physical motion of the particle with the energy imparted by the electric fields, illustrating the transformation of electrical energy into motion.

Voltage Acceleration Equivalence

The voltage acceleration equivalence is a concept that relates the kinetic energy gained by a charged particle to an equivalent voltage difference. This concept provides a straightforward comparison between the multistep acceleration in a cyclotron and a single potential jump, highlighting the energy required to accelerate a particle from rest to a given kinetic energy.

Incremental Energy Gain

Incremental energy gain refers to the energy a charged particle accumulates with each pass through the accelerating gap in a cyclotron. Since the particle only receives a small boost during every crossing, the cumulative effect over many revolutions leads to a significant total energy gain, making the process efficient for reaching high energies.

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