Q.Write the limitations of Cyclotron.

The Cyclotron: Why a Constant Frequency Can Accelerate a Particle to High Speeds

Imagine you want to throw a ball faster and faster, but you can only give it a small push each time. You could set up two paddles that slap the ball back and forth, each time adding a little speed. But the ball would just go in a straight line and fly away. To keep it contained, you need something to bend its path back toward you after each push.

That is the core idea of a cyclotron. It uses a magnetic field to bend the path of a charged particle into a circle, and an electric field (applied across two hollow D-shaped electrodes called "dees") to give it a kick of energy each time it crosses the gap between them. The trick is that the electric field must reverse direction at exactly the right moment — once per half-circle — so that it always pushes the particle forward, never backward.

The Surprising Fact: Frequency Does Not Depend on Speed

Here is the key insight that makes the cyclotron work. When a charged particle moves in a uniform magnetic field $B$, it experiences a centripetal force:

From this, the radius of its circular path is:

The time it takes to complete one full circle (the period $T$) is:

Notice: $v$ cancels out. The period — and therefore the frequency $f = 1/T$ — depends only on the charge $q$, the mass $m$, and the magnetic field $B$. It does not depend on how fast the particle is moving.

This is the cyclotron frequency. As the particle gains energy and its speed increases, its orbit radius grows (since $r = mv/qB$), but the time per revolution stays exactly the same. So you can set the alternating voltage across the dees to this fixed frequency, and it will always be in sync with the particle's motion — no matter how fast the particle gets.

How It Actually Works

1. A charged particle (say, a proton) is released near the centre, between the two dees.
2. A magnetic field perpendicular to the dees bends its path into a half-circle inside one dee.
3. When it reaches the gap, the electric field is oriented to accelerate it forward. The particle gains kinetic energy.
4. It enters the other dee with a slightly higher speed, so its next half-circle has a slightly larger radius.
5. By the time it returns to the gap, the electric field has reversed polarity — because exactly one half-period has passed — so it again accelerates the particle forward.
6. This repeats. Each crossing of the gap adds energy. The spiral path grows outward until the particle reaches the edge and is extracted.

The Limitation: Relativity …