KAHIBARO
Table of Contents
Basic idea
A cyclotron is a machine that accelerates charged particles by making them move in circles inside a magnetic field, while an alternating electric field gives them repeated pushes. The cyclotron principle is the central idea behind how this happens.
The key trick is simple. A magnetic field bends the path of a charged particle, but does not increase its speed. An electric field can increase the particle's speed. In a cyclotron, these two roles are separated. The magnetic field keeps the particle moving in a circular path, and the electric field speeds it up each time it crosses a gap.
Motion in a magnetic field
Consider a particle of charge $q$ and mass $m$ moving with speed $v$ perpendicular to a uniform magnetic field $B$. The magnetic force acts at right angles to the velocity, so it changes the direction of motion and produces circular motion.
The magnetic force has magnitude
$$
F_B = qvB
$$
For circular motion, this force acts as the centripetal force:
$$
qvB = \frac{mv^2}{r}
$$
Solving for the radius gives
$$
r = \frac{mv}{qB}
$$
This shows an important part of the cyclotron principle. As the particle gains speed, the radius of its circular path increases. So the particle spirals outward.
In a uniform magnetic field perpendicular to the motion,
$$
r = \frac{mv}{qB}
$$
A faster particle moves in a larger circle.
The role of the electric field
Inside the cyclotron are two hollow semicircular electrodes, often called dees because of their shape. There is a narrow gap between them. An alternating voltage is applied across this gap.
When the particle crosses the gap, the electric field in the gap accelerates it. Inside each dee, the electric field is essentially absent, so the particle is not accelerated there. It simply follows a semicircular path due to the magnetic field.
Each time the particle reaches the gap, the voltage must reverse so that the electric field pushes the particle forward again, not backward. In this way, the particle gains energy step by step.
Why repeated acceleration works
The particle takes half a circle inside one dee, then crosses the gap, then takes half a circle inside the other dee, then crosses again. If the electric field reverses in exactly the right rhythm, the particle gets accelerated every time it crosses the gap.
The cyclotron principle depends on the fact that, for non relativistic motion, the time taken for a half circle does not depend on the particle's speed.
From circular motion,
$$
v = r\omega
$$
and using $r = \frac{mv}{qB}$,
$$
v = \frac{mv}{qB}\omega
$$
which gives
$$
\omega = \frac{qB}{m}
$$
So the angular speed is constant. The period of one full revolution is
$$
T = \frac{2\pi}{\omega} = \frac{2\pi m}{qB}
$$
and the frequency is
$$
f = \frac{1}{T} = \frac{qB}{2\pi m}
$$
This frequency is called the cyclotron frequency.
Because $f$ does not depend on $v$ or $r$, the same alternating voltage can stay in step with the particle while the orbit gets larger.
For a non relativistic particle in a cyclotron,
$$
f = \frac{qB}{2\pi m}
$$
This is the basic reason repeated acceleration is possible. The orbital frequency is independent of the radius and speed.
Step by step picture
At the center, the particle starts with a small speed. The magnetic field bends it into a small semicircle. When it reaches the gap, the electric field accelerates it. Now it has more kinetic energy, so its speed is larger. The magnetic field still bends it, but now the radius is larger. It travels through the other dee in a bigger semicircle. When it crosses the gap again, the voltage has reversed, so it is accelerated once more. This process repeats many times, and the path becomes a spiral moving outward.
Energy gain at the gap
If the potential difference across the gap is $V$, then when the particle crosses the gap it gains kinetic energy
$$
\Delta K = qV
$$
If it crosses the gap many times, the total kinetic energy increases in steps. The particle is extracted when it reaches the outer edge with the desired energy.
When a charge $q$ crosses a potential difference $V$, the energy gained is
$$
\Delta K = qV
$$
The magnetic field bends the path, but the electric field provides the energy.
Geometry of the motion
The path is not one single circle of fixed radius. It is a sequence of semicircles with increasing radius. That is why the orbit looks like an outward spiral.
What makes the principle elegant
The elegance of the cyclotron principle is that one fixed magnetic field and one alternating voltage can accelerate a particle many times. The particle naturally returns to the gap after each half turn, and because the orbital frequency is constant in the non relativistic case, the electric field can be synchronized to keep pushing the particle forward.
This makes the device much more efficient than trying to accelerate the particle in one single push.
Conditions for the principle to work
The cyclotron principle works best when the particle speed is not too close to the speed of light. At low speeds, the mass can be treated as constant, and the frequency stays nearly constant. If the speed becomes very large, relativistic effects change the timing, and the particle no longer stays perfectly synchronized with the alternating field. That limitation belongs to later discussion of cyclotron limitations, but it is important to know that the simple principle assumes non relativistic motion.
Summary relations
The central ideas of the cyclotron principle can be collected in one place.
| Quantity | Relation | Meaning |
|---|---|---|
| Magnetic force | $F_B = qvB$ | Bends the path |
| Orbit radius | $r = \frac{mv}{qB}$ | Radius increases with speed |
| Angular speed | $\omega = \frac{qB}{m}$ | Constant for non relativistic motion |
| Cyclotron frequency | $f = \frac{qB}{2\pi m}$ | Frequency of revolution |
| Energy gain per gap crossing | $\Delta K = qV$ | Electric field increases kinetic energy |
The cyclotron principle combines two facts:
$$
r = \frac{mv}{qB}
$$
and
$$
f = \frac{qB}{2\pi m}
$$
So as the particle speeds up, the orbit gets larger, but the frequency stays the same, allowing repeated acceleration by an alternating electric field.