Name: Fleming's Left Hand Rule
Uploaded: 2019-07-19
Duration: 2 min 51 s
Description: In this video, we will look at using Fleming’s left-hand rule to help us remember how current, magnetic fields and direction go together. Get in to know more.

Question

Write the working of cyclotron in brief. Draw a schematic sketch of the cyclotron showing path of accelerated charged particles (ions) in both dees. Derive expression for cyclotron frequency.

Accepted Answer

Cyclotron

it is a device used to accelerate charged particles like protons, deuterons, α-particles, etc. to very high energies. Principle : A charged particle can be accelerated to high speeds (energies) by passing it through electric field many number of times and at the same time magnetic field makes the charged particle to move in a circular path.

Construction

A cyclotron consists of the following main parts

1. It consists of two small, hollow, metallic half-cylinders D1 and D2 , called dees (as they are in the shape of D).

2. They are mounted inside a vacuum chamber between the poles of a powerful electromagnet. 3. The dees are connected to the source of high frequency alternating voltage source of few hundred kilovolts.

4. The beam of charged particles to be accelerated is injected into the dees near their centre, in a plane perpendicular to the magnetic field.

5. The charged particles are pulled out of the dees by a deflecting plate (which is negatively charged) through a window W.

6. The whole device is in high vacuum (pressure – $10^{- 6}$ mm of Hg) so that the air molecules may not Collide with the charged particles. Theory : Let a particle of charge q and mass m enters a region of magnetic field vector B with a velocity v , normal to the field B . The particle follows a circular path, the necessary centripetal force is provided by the magnetic field.

Therefore, Magnetic force on charge q = Centripetal force on charge q or

qv B sin 90° = mv2/r or r = mv/qB

Period of revolution of the charged particle is given by-

$T = (\frac{2 π r}{v}) = (\frac{2 π}{v}) \frac{m v}{q B} = \frac{2 π m}{q B}$

Hence frequency of revolution of the particle will be,

$f_{c} = \frac{q B}{2 π m}$

Clearly, this frequency is independent of both the velocity of the particle and the radius of the orbit and is called cyclotron frequency or magnetic resonance frequency. This is the key fact which is used in the operation of a cyclotron.

Working

Suppose a positive ion, say a proton, enters the gap between the two dees and finds dee D1 to be negative. It gets accelerated towards dee D1 .

As it enters the dee D1, it does not experience any electric field due to shielding effect of the metallic dee. The perpendicular magnetic field throws it into a circular path with constant speed. At the instant the proton comes out of dee D1, it finds dee D1 positive and dee D2 negative. It now gets accelerated towards dee D2.

• It moves faster through D2 describing a larger semicircle than before.

• Thus if the frequency of the applied voltage is kept exactly the same as the frequency of revolution of the proton, then every time the proton reaches the gap between the two dees, the electric field is reversed and proton receives a push and finally it acquires very high energy

. • This condition in which frequency of applied voltage is equal to the frequency of revolution of charged particle is called the cyclotron’s resonance condition.

• The accelerated proton is ejected through a window by a deflecting voltage and hits the target. Maximum K.E. of the accelerated ions : The ions will attain maximum velocity near the periphery of the dees. If v0 is the maximum velocity acquired by the ions and r0 is the radius of the dees, then,

$\frac{m {v_{0}}^{2}}{r_{0}} = q v_{0} B$

The maximum kinetic energy of the ions will be

$K_{0} = \frac{1}{2} m {v_{0}}^{2} o r K_{0} = \frac{q^{2} B^{2} {r_{0}}^{2}}{2 m}$

frequency of revolution of the particle will be,

$f_{c} = \frac{1}{T} = \frac{q B}{2 π m}$

Write the working of cyclotron in brief. Draw a schematic sketch of the cyclotron showing path of accelerated charged particles (ions) in both dees. Derive expression for cyclotron frequency.

Important Questions on Moving Charges and Magnetism

Learn and Prepare for any exam you want !