Question

A cyclotron is used to accelerate charged particles. Then the time period under the influence of $1T$ magnetic field of a proton:
$\begin{aligned} & \text{A}\text{. }20\pi \text{ }ns \\\ & \text{B}\text{. }40\pi \text{ }ns\text{ } \\\ & \text{C}\text{. }10\pi \text{ }ns \\\ & \text{D}\text{. }5\pi \text{ }ns \\\ \end{aligned}$

Answer 1

A charged particle or ion can acquire sufficiently large energy with a comparatively smaller alternating potential difference by making them cross the same electric field again and again with the help of a strong magnetic field. Time period of the particle in motion between the charged plates in the cyclotron can be calculated by using the relationship between the time period, the magnitude of the magnetic field and the charge by mass ratio of the particle.

Formula used:
Time period of a charged particle in a cyclotron,
$T=\dfrac{2\pi m}{qB}$

Complete step by step answer:
A cyclotron is a machine that is used to accelerate the charged particles or ions to high energies. Both the electric field and the magnetic field are used in a cyclotron in order to increase the energy of the charged particles. As both the fields are perpendicular to each other, they are known as the cross fields.
In a cyclotron, charged particles accelerate in the outward direction from the centre along a spiral path. These particles are held to a spiral trajectory by a static magnetic field and accelerated with the help of a rapidly varying electric field.
Cyclotron works on the principle that a charged particle or an ion moving in a direction normal, or perpendicular, to the existing magnetic field experiences magnetic Lorentz force as a result of which the particle moves in a circular path.

The purpose of the electrodes placed in the set-up is to accelerate the charged particle. The purpose of the magnetic field is to bend the particle round so that it goes back towards the other electrode. In a cyclotron, the electric field makes the particle accelerate when it is in between the plates.
Time period of a charged particle in a cyclotron is given by,
$T=\dfrac{2\pi m}{qB}$
Where,
$m$ is the mass of the charged particle
$q$ is the magnitude of charge
$B$ is the magnitude of magnetic field
$T=\dfrac{2\pi }{B}\left( \dfrac{m}{q} \right)$
$\dfrac{q}{m}$ is defined as the charge by mass ratio of a particle
Putting values,
$\begin{aligned} & B=1\text{ }T \\\ & \dfrac{q}{m}=9.6\times {{10}^{7}}\text{ }Ck{{g}^{-1}} \\\ \end{aligned}$
(Charge by mass ratio of a proton is $9.6\times {{10}^{7}}\text{ }Ck{{g}^{-1}}$)
We get,
$\begin{aligned} & T=\dfrac{2\pi }{1}\left( \dfrac{1}{9.6\times {{10}^{7}}} \right) \\\ & T=\dfrac{2\pi }{9.6\times {{10}^{7}}}=20\pi \times {{10}^{-9}} \\\ & T=20\pi \text{ }ns \\\ \end{aligned}$
The time period of the proton is $20\pi \text{ }ns$
Hence, the correct option is A.

Note:
There are some limitations to the Cyclotron:
A cyclotron cannot accelerate electrons because electrons have very small mass.
A cyclotron cannot be used to accelerate neutral particles.
A cyclotron cannot accelerate positively charged particles with large mass due to the relativistic effect.