A proton and an alpha particle are accelerated in a cyclotron. For the exit circular path of radius r, their velocities are v_H and v_α respectively, then

CONCEPT:

• A cyclotron is a machine that is used to accelerate charged particles or ions to high energies.
• A simple cyclotron consists of a large circular magnet providing a constant magnetic field across the gap between the pole-faces.
• A charged particle is injected in the middle of the magnetic field and the particle is accelerated in the magnetic field in a circular shape.
• The radius of the circular path (R) is given by:
  $R=\frac{mv}{qB}$
• Angular velocity (ω) is given by:
  $\omega =\frac{Bq}{m}$
  Where B = magnetic field in the cyclotron, q = charge of particle, m = mass of the particle, and v = velocity of particle.

EXPLANATION:

• The radius of the circular path (R) is given by:
  $\Rightarrow R=\frac{mv}{qB}$
  The above equation can be written for velocity as
  $\Rightarrow v=\frac{qBR}{m}$
  For proton, the velocity will be
  $\Rightarrow v_H=\frac{1\times Br}{1}=Br$
  For alpha particle, the velocity will be
  $\Rightarrow v_{\alpha }=\frac{2\times Br}{4}=\frac{Br}{2}$
  ⇒ vH > vα
• Therefore option 1 is correct.