Question

What is a cyclotron? State its two limitations.

Hint:

A cyclotron accelerates charged particles using perpendicular electric and magnetic fields. It is mainly used in nuclear physics and medical applications (like particle therapy).

Answer 1

Step 1: In a cyclotron, charged particles (such as protons or ions) are injected at the center between two hollow semi-circular electrodes called \emph{dees}. These particles are accelerated each time they cross the gap between the dees by an alternating electric field.
Step 2: A strong magnetic field \( \mathbf{B} \), applied perpendicular to the plane of the dees, forces the charged particles to move in a circular path due to the Lorentz force. As the particles gain energy and speed with each acceleration, the radius of their circular trajectory increases.
Step 3: The frequency \( f \) of revolution of the charged particles in the magnetic field is given by: \[ f = \frac{qB}{2\pi m}, \] where \( q \) is the charge of the particle, \( B \) is the magnetic field strength, and \( m \) is the mass of the particle. This frequency, known as the cyclotron frequency, remains constant for non-relativistic speeds.
Step 4: However, for very light particles like electrons, when their speeds approach a significant fraction of the speed of light, relativistic effects become important. The effective mass of the particle increases, which reduces the frequency of revolution. This causes the particle to fall out of sync with the alternating electric field, leading to synchronization failure and limiting the cyclotron's ability to accelerate electrons efficiently.