10.0 Cyclotron

  Magnetics

Cyclotron is a device which is used to accelerate positive particles like $α-$particle, deuteron etc. It is based on the fact that the electric field accelerates a charged particle and the magnetic field keeps it revolving in circular orbits of increasing radius. It consists of two hollow $D-$shaped metallic chambers $D1$ and $D2$ called dees. The dees are connected to the source of high-frequency electric field. The whole apparatus is placed between the two poles of a strong electromagnet $N–S$ as shown in. The magnetic field acts perpendicular to the plane of the dees

$i.$ Cyclotron frequency$:-$ Time taken by charged particle to describe a semicircular path,$$t = \frac{{\pi r}}{v} = \frac{{\pi m}}{{qB}}$$ The period of oscillating electric field$$T = 2t = \frac{{2\pi m}}{{qB}}$$ and cyclotron frequency$$f = \frac{1}{T} = \frac{{Bq}}{{2\pi m}}$$$ii.$ Maximum kinetic energy of the particle $:-$

(a) We have $$r = \frac{{mv}}{{qB}}$$for$${r_0} = \frac{{mv}}{{qB}}$$$$ \Rightarrow \quad {v_0} = \frac{{{r_0}qB}}{m}$$where $(r_0$ maximum radius of circular path$)$$$K.E = \frac{1}{2}mv_0^2 = \frac{1}{2}m{\left( {\frac{{{r_0}qB}}{m}} \right)^2} = \left( {\frac{{{q^2}{B^2}}}{{2m}}} \right)r_0^2$$(b) Also, $K.E.\ =$ work done by electric sources $:-$$$K.E = f \times \left( {qV \times 2} \right)$$$$K.E = 2f\;qV$$