Question

With the help of a labelled diagram, state the underlying principle of a cyclotron. Explain clearly how it works to accelerate the charged particles. Show that cyclotron frequency is independent of energy of the particle. Is there an upper limit on the energy acquired by the particle? Give reason.

Answer 1

Principle: It is based upon the principle that a charged particle moving at right angles to a uniform magnetic field is acted upon by a force perpendicular to its direction of motion and follow a circular path.

Construction: It consists of two semicircular hollow metal boxes D₁ and D₂ called dees. D's are separated by a small gap and are connected to terminals of high frequency oscillator providing alternating potential of nearly 10,000 V and millions hertz frequency. Whole of the apparatus is placed in a metal box containing gas at low pressure and placed between the poles of a strong electromagnetic field perpendicular to the plane of D's.

Working

Let a positive ion (₁H¹, ₁H², ₂He³ or ₂He⁴) be located in the gap when D₁ is negatively and D₂ positively charged, the positive ion will get accelerated towards D₁. Since magnetic field is uniform and acting at right angle to the plane of D’s, the ions traverse a circular path in When the ion emerges out into the gap after completing a semicircle in D₁. If during the time taken to cover semicircle, the electric field gets reversed, the ions further get accelerated towards D₂. It enters D₂ with greater speed and moves in D₂ in a bigger semicircle. The process is repeated time and again and each time ions get an additional kick in the proper direction. The ion becomes faster and faster until it reaches the periphery of the D's where it is brought out of the chamber by means of deflecting plate P charged to a very high negative potential and is made for bombarding the target.

Theory

Let m = mass of the ion to be accelerated,

B = flux density of the magnetic field,

r = radius of the semicircular path in which the ion moves,

q = charge on the ion,

v = velocity with which the ion moves in the semicircle 

Force acting on the charged particle

F = Bqv sin 90°

or F = B qv

This force provides the necessary centripetal force because it moves along circular path

∴ \(\frac{mv^2}{r}\) = Bqv

∴ V = \(\frac{Bqr}{m}\)

Time taken by charged particle to complete half circle,

t = \(\frac{\text { half circumference }}{\text { velocity }} = \frac{\pi r}{v}\)

= \(\frac{\pi r}{Bqr/m}\) = \(\frac{\pi m}{Bqr}\)

As m, B, q are constant,

∴ t is constant.

This clearly shows that time taken by the charged particle to complete a semicircle is independent of its speed and the radius of circle in which it moves.

Let T = time period of oscillator

∴ T = 2t = \(\frac{2\pi m}{Bq}\)

Frequency of oscillator field

V = \(\frac{1}{T}\) = \(\frac{Bq}{2\pi m}\)

This frequency is constant and is independent of velocity (or energy) of the particle and radius of the circular path in which it moves.

Limitations.

1. The cyclotron is unable to accelerate the ions beyond a certain limit since mass will increase according to equation m = \(\frac{m_0}{\sqrt {1-\frac{v^2}{e^2}}}\), whenever v²/c² are not negligible.

Hence required frequency will now be

v = \(\frac{Bq\sqrt{1-v^2/c^2}}{2\pi m_0}\)

Since we are unable to change the frequency in short duration, hence ion will get out of phase with high frequency accelerating field and thus lose energy instead of gaining.

2. Electrons cannot be accelerated with cyclotron.

3. Neutrons cannot be accelerated with cyclotron.

A rotating charged particle is always accelerated and radiate electromagnetic energy.