Question

In a cyclotron as radius of the circular path of the charged particle increases ($\omega $= angular velocity, v = linear velocity).
A. Both $\omega $ and v increases
B.$\omega $ only increases, v remains constant
C. v increases, $\omega $ remains constant
D. v increases, $\omega $ decreases

Answer 1

Hint: Cyclotron is a machine which is used to accelerate the positive charges by using both the electrical and magnetic field rapidly on particle that is travelling outside in spiral path now in order to solution of above question we need to gain formula the velocity by equalizing centripetal force and Lorentz force.
Formula used:
 ${{F}_{c}}=\dfrac{m{{v}^{2}}}{r}$
And
${{F}_{B}}=qvB$

Complete answer:
In order to keep particles in a curved path, centripetal force and Lorentz’s force on the magnetic field need to be equal.
Centripetal force = Lorentz’s force
$\begin{align}
  & \Rightarrow {{F}_{c}}={{F}_{B}} \\
 & \therefore \dfrac{m{{v}^{2}}}{r}=qvB \\
\end{align}$
Where,
${{F}_{c}}$ = Centripetal force
m = mass
v = velocity
r = radius
${{F}_{B}}=$ Lorentz’s force
q = charge
v = velocity
B = magnetic field
$\begin{align}
  & \Rightarrow v=\dfrac{qBr}{m}......\left( 1 \right) \\
 & \therefore v\propto r......\left( 2 \right) \\
\end{align}$
Now formula for angular velocity
$\omega =\dfrac{v}{r}$
Now from the equation (1)
$\begin{align}
  & \Rightarrow \dfrac{v}{r}=\dfrac{qB}{m} \\
 & \therefore \omega =\dfrac{qB}{m}.....\left( 3 \right) \\
\end{align}$
So from the equation (2) we can state that velocity will increase as the radius of the circular path of the charged particle increases.
And from equation (3) we can use the statement that angular velocity will be constant when the radius at the circular path at the charged particle increases.

Hence the correct option is (C).

Additional information:
$\to $ Cyclotron is the machine which is used in many useful applications. Below is a common application in which cyclotron is used.
$\to $ Its main application is to accelerate any charged particle mainly positive charges it is used in physics and also in medicine
$\to $ It produces radioactive isotopes and these radioactive isotopes are used in nuclear medicines and some other useful applications.
$\to $ Cyclotron has a very useful application and that is used to cure the cancer patients.
$\to $ When a particle has more or less velocity then we can use a cyclotron to gain the required amount of velocity.

Note:
If we want a direct relation between angular velocity and the required formula, then we can directly remember the formula $v=\dfrac{qBr}{m}$ for velocity so that we can solve this question very fast.