Question

In cyclotron, an ion is made to travel along semicircles of increasing radii under a magnetic field. The angular velocity of this ion is independent of
A) speed of ion
B) radius of the circle
C) mass of the ion
D) charge of the ion

Answer 1

Let us first find the net forces acting on the particle when it is travelling along the semicircles of increasing radii. Next, as the net forces acting on the system are zero and the net forces are magnetic force and centripetal force, they both must be equal. Find the angular velocity in terms of possible parameters and verify.

Formula used:
$\begin{aligned} & {{F}_{c}}=\dfrac{m{{v}^{2}}}{r} \\\ & {{F}_{m}}=qvB \\\ \end{aligned}$

Complete step-by-step answer:
Let us find out the forces acting on the particle. The two possible forces are the centripetal force of the ion as it is moving in a circular shape. Also, it is given that the ion is successively moving because of the presence of the magnetic fields. Therefore, the magnetic field force is also acting on the particle. As there are no other forces, these two forces must be equal to each other, which is,
$\begin{aligned} & \Rightarrow {{F}_{c}}=\dfrac{m{{v}^{2}}}{r} \\\ & \Rightarrow {{F}_{m}}=qvB \\\ & \Rightarrow {{F}_{c}}={{F}_{m}} \\\ & \Rightarrow \dfrac{m{{v}^{2}}}{r}=qvB \\\ & \Rightarrow v=\dfrac{qBr}{m} \\\ & (v=r\omega ) \\\ & \Rightarrow \omega =\dfrac{qB}{m} \\\ \end{aligned}$
We can clearly see, the angular velocity of the ion is only dependent on the charge of the ion, magnetic field on the ion and the mass of the ion. Therefore, the angular velocity is independent of the speed of the ion and radius of the semicircle.

So, the correct answer is “Option A and B”.

Additional Information: A cyclotron is a typical particle accelerator invented by Ernest o. Lawrence at the university of California. Its task is to accelerate the charged particles outwards from the centre of a flat cylindrical vacuum chamber in a spiral path or simply in a circular way. Centripetal force is the force acting on a body towards the centre when a body moves in a circular path.

Note: A charged particle always follows a curved path in a magnetic field because the magnetic force acting on it is always perpendicular to the direction of travel. In a vacuum, the magnetic field will be the dominant factor as there’s no other force acting. A particle charged in a magnetic field can never be stopped but can be trapped by making it do circles around a centre. It has been proved that particles charged in a magnetic field can escape into infinity without ever stopping.