# Investigatory Project on Cyclotron

**Investigatory Project on Cyclotron for Class 12th**

**Aim:** To accelerate positively charged particles like
proton, deuteron etc.

**Principle: **

Positive ions are subjected to a magnetic field perpendicular to the direction of the velocity of the ions. The magnetic force Bqv acting on the ions keeps them in a circular path in two evacuated hollow semicircular dees. The radius of the orbit increases as ions are accelerated by an alternating voltage applied between the dees. The governing equation is,

mv^{2}/r = Bqv, where m is the mass of the ion and r
the radius of the circular orbit.

**Materials and Apparatus: **

It consists of two semicircular flat metal (ex. copper)
boxes D_{1} and D_{2} called `dees' and are constructed with a
small gap amid them. A source of ions is positioned near the centre point of
the gap amid the `dees'. The `dees' are coupled to the terminals of a radio frequency
oscillator, such that the high frequency alternating potential having several
MHz resides amid the dees, which work as electrodes. The `dees' are included in
insulated metal (such as copper) which is well evacuated. The entire apparatus
is located amid the pole pieces of a strong electromagnet that offers a
magnetic field perpendicular to the flat plane of the `dees'.

**Working and theory **

The positive ion to be accelerated is produced by the source
s. Suppose at this instant D_{1} is at negative potential and D_{2}
at positive potential. Therefore the ions will be accelerated towards D_{1}.
On entering D_{1} it is affected by the perpendicular magnetic field
and hence it describes a semicircular path of radius r such that,

mv^{2}/r = Bqv

Therefore, r = mv/Bq

Time taken by the ion to describe the semicircular path is given by,

t = πr/v = πm/Bq = a constant

Thus the time is independent of both speed of the ion and
radius of the circular path. The frequency of the oscillator is so adjusted
that the time taken by the ion to describe a semicircular path is equal to half
period of the oscillator. Thus when the ion reaches the gap the polarity of the
dees is reversed. i.e., D_{1} becomes positive and D_{2 }negative.
The positive ion is accelerated towards D_{2}. It describes a circular
path of larger radius and will arrive at the gap just when the polarity of the
dees is reversed. Thus the positive ion goes on accelerating every time it
reaches the gap between the dees and finally acquires the required kinetic
energy. The accelerated ion can be deflected out of the dee to the target
through a window by a deflecting electrode.

**Maximum energy of positive ion **

If R is the maximum radius of the circular path which is
almost equal to the radius of the 'dee' and v_{max} the maximum
velocity of ion, then,

Mv_{max}^{2}/R = Bqv_{max}

Therefore, v_{max} = BqR/m

Maximum kinetic energy = ½mv_{ max}^{2}
= ½m
x (BqR/m)^{2} = B^{2}q^{2}R^{2}/2m

**Cyclotron frequency **

If T is the time period of the oscillator, then,

T = 2t = 2πm/Bq

The cyclotron frequency, f = (Bq/2πm). This is known as magnetic resonance frequency. This is the same as the frequency of the oscillator.

**Limitations of cyclotron **

1. If the positive ion is accelerated to very high speed
comparable to the speed of light, its mass varies according to the relation, m
= m_{0} /√([1 - v^{2}]/c^{2}; where m_{0} is
the rest mass of the ion. Hence the equation for frequency changes. So the ion
cannot be accelerated beyond a certain limit.

2. Cyclotron is suitable only for accelerating heavy particles. Electrons cannot be accelerated by cyclotron.

3. Uncharged particles also cannot be accelerated by cyclotron.

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