Question

Assertion: Cyclotron is a device which is used to accelerate the positive ions
Reason: Cyclotron frequency depends upon the velocity.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.

Answer 1

Cyclotron is a device used to accelerate heavy charged particles such as protons, deuterons, $\alpha$-particles. In the cyclotron, a charged particle acquires kinetic energy by the help of an alternating electric field and moves in a circular path with the help of a strong magnetic field. The frequency and time period of a cyclotron is independent of both the speed of the ion and radius of the circular path.

Complete step by step answer:
A cyclotron employs a uniform magnetic field and a uniform electric field. It is based on the principle that a charged particle can be accelerated to very high energy with a smaller alternating electric field by making it cross the same electric field time and again with the help of a strong magnetic field. Since the electric and magnetic fields are perpendicular to each other, the ion moves in a circular path. Cyclotron used to accelerate heavily charged particles such as protons, deuterons, $\alpha$-particles. Suppose a positive ion $+ q$ of mass $m$ is placed at the center of a cyclotron. The strength of the magnetic field is $B$.

Let’s at any time, the radius of the circular path is $r$ and the speed of the positive ion is $v$. Then, the magnetic force on the charge is equal to the centrifugal force on the charge.
$qvB = \dfrac{{m{v^2}}}{r}$
Radius of path, $r = \dfrac{{mv}}{{qB}}$
The time period of the positive ion i.e., time taken to complete circle of radius $r$ is
$T = \dfrac{{2\pi r}}{v}$
Substitute the value of $r$ in the above formula
$T = \dfrac{{2\pi }}{v}\left( {\dfrac{{mv}}{{qB}}} \right)$
$\Rightarrow T = \dfrac{{2\pi m}}{{qB}}$
Now cyclotron frequency, $f = \dfrac{1}{T}$
$\therefore f = \dfrac{{qB}}{{2\pi m}}$
Clearly, the cyclotron frequency is independent of both the speed $\left( v \right)$ of the ion and radius $\left( r \right)$ of the circular path.

Hence, the correct option is C.

Note: There are some limitations of cyclotron which are discussed below:
- Cyclotron is not suitable for accelerating electrons. It is because the mass of electrons is very small and hence gains speed quickly due to small increase in energy. As a result, the relativistic variation of mass makes electrons out of step with the oscillating electric field.
- Cyclotron cannot accelerate an uncharged particle such as a neutron.