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Principle of a cyclotron - Cyclotron is a device that works on the principle that the kinetic energy of a charged particle increases when the charged particle moves in a crossed oscillating electric and magnetic field
It's working ;
When a positive charge q of mass m is emitted from the source, it is accelerated towards the Dee having a negative potential, due to normal magnetic field, the ions experience a magnetic force and moves in a circular path. When the charged particle arrives at the gap between the dee the polarities of the Dee changes and in the gap the charged particles gain extra velocity and is then transferred to the another Dee making a circular path with the greater radius.
In such a way the charged particles keep moving in the two Dees each time increasing in velocity and radius of path. When it reaches its maximum radius the charged particle comes out of the Dee with high velocity and hits the deflection plate T.
Let us consider the charge on the positive ion is +q, its mass m and strength of the magnetic field is B.
At any time, 'r' is the radius of the circular path and 'v' be the speed of the particle. Then, on equating the centripetal force and Lorentz's force, we have
Now, to complete the circle of 'r' the time taken is given by -
on substituting the value of r, we get
Since,
Then,
Hence, the required expression is -