Cyclotron

Introduction: Cyclotron

Cyclotron is a device used to accelerate charged particles to a very high speed, that is, to a very high kinetic energy.

These high speed charged particles are used in the following purpose:

• To study nuclear reactions

• To insert an ion in a solid

• Used to start radioactive decay

So for all these purposes, Cyclotron is used.

Since we need high velocity charged particles in this device so we will keep the charged particles between two oppositely charged plates. The electric field increases speed because it exerts an accelerating electric force on the charged particles when kept between two oppositely charged plates as shown in figure 1.

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Figure 1

Here, $\mathrm{acceleration(a) \:=\:\frac{Force(F)}{Mass(m)}}$ and the Force(F) is nothing but the product of charge(q) and electric field(E).

Here if we want to increase the velocity more then there are two ways as demonstrated below -

We can increase the distance between the plates

$\mathrm{V^2\:=\:u^2\:+\:2.a.s}$

Here, v = final velocity of the charged particle

u = initial velocity of the charged particle

a = acceleration

s = distance traveled

So, the final velocity is directly depending on the distance traveled.

We can put the charge in the electric field for more time

v = u + a.t

here, t = time

So, the final velocity is also directly depending on the time.

Now if we increase ‘s’ then the size of the device will increase which is not desirable.

Also, Electric field intensity is inversely proportional to ‘s’ so it will also decrease and that will decrease the acceleration. So we will go for the second option that is, increasing the time. It can be possible if we introduce magnetic fields also in the device. The magnetic field will change the direction of the charged particle and start rotating the charged particle between the charged plates which will increase the time ‘t’ and hence the speed of the charged particle (v) will increase. So here we introduce cross field that is, direction of electric field and magnetic field will be perpendicular to each other.

Furthermore, let us see the construction of the Cyclotron.

Construction of Cyclotron

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Figure - 2(a)

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Figure - 2(b)

Here, $\mathrm{D_1}$ and $\mathrm{D_2}$ are hollow metallic semi-circular containers known as Dees [figure 2 (a)]. Also, the source of charged particles has a very high frequency (1 lakh to 100 lac Hertz), since the frequency is so high that is why it is also called Oscillator [figure 2 (b)]. The direction of the magnetic field is outside of the plane of paper [figure 2 (a)]. The voltage of the source is very high so it will increase the electric field and so will the electric force.

Working of Cyclotron

Let a charge ‘q’ from the source is emitted and accelerated towards $\mathrm{D_2}$ (negatively charged at that moment) [figure 2 (b)], then its speed increases. As soon as ‘q’ enters $\mathrm{D_2}$, electric field intensity(E) = 0 (because E inside a conducting hollow body is zero).

Now its speed cannot increase but the magnetic field will force ‘q’ to move in a circular path with the same speed. As ‘q’ comes out from $\mathrm{D_2}$ then the polarity of Dees switches ($\mathrm{D_1}$ will become negative and $\mathrm{D_2}$ will become positive). Now ‘q’ accelerates towards $\mathrm{D_1}$ and as it enters $\mathrm{D_1}$ again E = 0 and the magnetic field forces it to move in a circular path with the same speed. But now the radius of the circular path is more than the previous one. This happens because during movement from $\mathrm{D_2}$ to $\mathrm{D_1}$, it accelerates and its velocity increases.

And as we know that in a magnetic field -

$$\mathrm{r\:=\:\frac{m.v}{q.B}}$$

Here, r = radius of the circular path, m = mass of the charged particle, v = velocity of the charged particle, q = charge on the charged particle, B = Magnitude of the magnetic field. So as the velocity of the charged particle increases, the radius of the circular path increases.

So from the figure 2 (a) and 2 (b), we can see that the radius of the circular path is increasing and we are getting a spiral path.

Final Speed and Kinetic Energy

When the charged particle exits out, the radius of the circular path will be equal to the radius of Dees. Also we know that,

$$\mathrm{r\:=\:\frac{m.v}{q.B}}$$

So, we can write the velocity of charged particle as,

$$\mathrm{v\:=\:\frac{qB.r}{m}}$$

Also we know that the kinetic energy(K.E.) can be written as –

$$\mathrm{K.E. \:=\:\frac{1}{2}m.v^2\:=\:\frac{1}{2}m.(\frac{qB.r}{m})^2\:=\:\frac{q^2B^2r^2}{2m}}$$

Coming to the time period(T) of the charged particle ‘q’ for circular motion -

$$\mathrm{T\:=\:\frac{2\pi m}{qB}}$$

As we know that the frequency(f) will be reciprocal of the time period, so it can be written as -

$$\mathrm{f\:=\:\frac{qB}{2\pi m}}$$

This is also called the Oscillator frequency or Cyclotron frequency.

Limitations of Cyclotron

Some of the limitations of Cyclotron are as mentioned below –

• It cannot be used for neutral particles like neutrons because neutral particles cannot accelerate.

• It cannot be used for very small charged particles like electrons. Because the mass of the electron is very less, almost 10000 times smaller than protons. The reason is that since mass is very less so the driving acceleration will be very high so the velocity will be very high. Because of that, the velocity of electrons becomes comparable with the speed of light due to which the mass of the electron will increase in each revolution. Due to this frequency of the charged particle will decrease but the frequency of the Oscillator is fixed, so there will be a mismatch between charged particle and oscillator frequency. So the electron falls out of step from the cyclotron frequency. And we can also say that the Cyclotron cannot accelerate any charged particle whose speed is comparable to the speed of light.

FAQs

Q1. What is a Cyclotron?

Ans. It is a device used to accelerate the charged particle to a very high speed. The accelerated charged particles are useful in many purposes.

Q2. Why magnetic fields are necessary for the Cyclotron along with electric fields?

Ans. Electric fields are used to accelerate the charged particles but magnetic fields change the direction of motion. It gives circular motion to the charged particle so that it will spend more time in the electric field and it can gain very high velocity.

Q3. Give some application of Cyclotron.

Ans. Cyclotrons are used in -

• In the manufacturing of medicines related to disease like Cancer

• To insert an ion in a solid.

Q4. Why is cyclotron not used to accelerate neutrons?

Ans. Neutron is a neutral particle and it will not get accelerated by the electric field. So its velocity will not be increased in the Cyclotron.

Q5. Why is the Cyclotron not used to accelerate the electrons? Please specify the exact reason in brief.

Ans. The mass of the electron is very less. It is almost 10000 times smaller than that of the proton. Due to this, there will be a mismatch in the frequency of the charged particle and the cyclotron.

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