The cyclotron is a device for accelerating protons (or other...

The cyclotron is a device for accelerating protons (or other charged particles) to high energies. The protons are held in a circular orbit of radius $R=p / e B$ by a uniform magnetic field $B .$ Twice in each orbit they are subjected to an accelerating electric field. As they speed up, $R$ increases and they spiral slowly outward. (a) Show that the period of each orbit is $T=2 \pi \gamma m /(e B)$. (b) As long as the motion is nonrelativistic, $\gamma \approx 1$ and the period is constant. This greatly simplifies the design of the cyclotron, since the accelerating field can be applied at a constant frequency. Assuming that a cyclotron can tolerate no more than a $2 \%$ increase in the period, what is the highest kinetic energy of the protons it can produce?

Key Concepts

Cyclotron Operation

A cyclotron accelerates charged particles using a combination of a static magnetic field, which forces the particles into a circular path, and an oscillating electric field, which accelerates the particles each time they cross the gap between the dees. The synchronization between the particle’s circular motion and the applied electric field is key to the device’s operation.

Lorentz Force and Circular Motion

Charged particles moving in a magnetic field experience a Lorentz force that acts perpendicular to their velocity, causing them to move in circular trajectories. The balance between the magnetic force and the centripetal force results in the radius of motion being proportional to the particle’s momentum and inversely proportional to the magnetic field strength.

Relativistic Effects

As a charged particle approaches high speeds, relativistic effects become significant, manifesting in an increase in the particle’s effective mass, as described by the gamma factor (?). This increase alters the cyclotron’s frequency and period, complicating the synchronization of the electric field with the particle’s motion.

Time Dilation and Period of Circular Motion

Incorporating relativistic effects leads to a modification of the particle’s orbital period. The period of motion in a cyclotron, expressed as T = 2??m/(eB), shows that as the velocity approaches relativistic speeds (? > 1), each orbit takes longer due to time dilation, thereby affecting the efficiency of acceleration.

Nonrelativistic Approximation

In the low-speed (nonrelativistic) regime, the gamma factor is approximately equal to 1. This simplifies the cyclotron’s design because the orbital period remains nearly constant, allowing the constant-frequency electric field to effectively accelerate the particles.

Design Limitations Due to Period Variation

The practical design of cyclotrons involves constraints to ensure that the synchronization between the particle motion and the electric field is maintained. A small allowable variation in the period, typically on the order of a few percent, limits the maximum kinetic energy that can be imparted to the particles without significant loss of efficiency.

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