Question

An α-particle and a proton are released from the centre of the cyclotron and made to accelerate.

(i) Can both be accelerated at the same cyclotron frequency?

Give reason to justify your answer.

(ii) When they are accelerated in turn, which of the two will have higher velocity at the exit slit of the does?

Answer 1

(i) Let us consider: Mass of proton = m, Charge of proton = q, Mass of alpha particle = 4m

Charge of alpha particle = 2q

Cyclotron frequency, ${\displaystyle v=\frac{Bq}{2\pi m}\Rightarrow v\propto \frac{q}{m}}$

For proton : Frequency , ${\displaystyle v_p\propto \frac{q}{m}}$

For alpha particle: Frequency ,${\displaystyle v_a\propto \frac{2q}{4m}\text{or}{v}_a\propto \frac{q}{2m}}$

Thus, particles will not accelerate with same cyclotron frequency. The frequency of proton is twice than the frequency of alpha particle.

(ii)

Velocity,${\displaystyle v=\frac{Bqr}{m}\Rightarrow v\propto \frac{q}{m}}$

for proton:Velocity, ${\displaystyle v_p\propto \frac{q}{m}}$

For alpha particle : Velocity ,${\displaystyle v_a\propto \frac{2q}{4m}\text{or}{v}_a\propto \frac{q}{2m}}$

Thus particles will not exit the dees with same velocity. The velocity of proton is twice than the velocity of alpha particle.