The magnetic field in a cyclotron used to produce...

The magnetic field in a cyclotron used to produce radioactive tracers is $0.50 \mathrm{T}$. What must be the mini-
mum radius of the dees if the maximum proton speed desired is $1.0 \times 10^{7} \mathrm{m} / \mathrm{s} ?$

Key Concepts

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge of the particle, v is its velocity, and B is the magnetic field. This force is always perpendicular to the direction of motion, which causes the particle to follow a curved or circular trajectory in the presence of a uniform magnetic field.

Radius of Circular Motion

When a charged particle moves perpendicular to a uniform magnetic field, the Lorentz force provides the necessary centripetal force to keep the particle in circular motion. The radius of the path is determined by balancing the magnetic force (qBv) with the centripetal force (mv²/r), leading to the formula r = mv / (qB). This concept is essential in understanding how the size of the circular path depends on the particle’s mass, speed, charge, and the strength of the magnetic field.

Cyclotron Operation

A cyclotron is a type of particle accelerator that uses a perpendicular magnetic field and an oscillating electric field to accelerate charged particles in a spiral path. The magnetic field causes the particles to follow circular trajectories inside large, semicircular electrodes called dees. The dees are designed so that the particles experience an accelerating potential each time they cross the gap between them, thus gaining energy and increasing in speed until they reach a desired energy level.

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