Question

Maximum kinetic energy of a positive ion in a cyclotron is
A. $\dfrac{{{q}^{2}}B{{r}_{0}}}{2m}$
B. $\dfrac{q{{B}^{2}}{{r}_{0}}}{2m}$
C. $\dfrac{{{q}^{2}}{{B}^{2}}r_{0}^{2}}{2m}$
D. $\dfrac{qBr}{2{{m}^{2}}}$

Answer 1

Hint: Equate the force due to magnetic field and the centrifugal force of the positive ion. Velocity is the subject. Then the value is substituted in the kinetic energy equation. On rearranging the equation we get the required answer. Formula used for this solution is
$q{{v}_{0}}B=\dfrac{mv_{0}^{2}}{{{r}_{0}}}$

Complete step by step answer:
A cyclotron is a particle accelerator. This machine is provided electricity to produce a beam of charged particles which can be used in different fields of work like research purposes and medical purposes. First drawback is that it cannot accelerate neutral particles. Second drawback is that the subatomic particles with lighter mass like electrons or positrons cannot be accelerated. They require important improvements to the device.

In this graph, we see the first cyclotron resonance which leads to an increase in amplitude.
In a cyclotron, force by a magnetic field is equal to the centrifugal force,
$\therefore q{{v}_{0}}B=\dfrac{mv_{0}^{2}}{{{r}_{0}}}$ where $r_0$ is maximum radius of the circular path of positive ion.
${{v}_{0}}$ is the maximum speed of positive ions.
Let us make $v_0$ as the subject and we get
${{v}_{0}}=\dfrac{qB{{r}_{0}}}{m}$……. (1)

Maximum kinetic energy of ion=$\dfrac{1}{2}mv_{0}^{2}=\dfrac{1}{2}m{{(\dfrac{qB{{r}_{0}}}{m})}^{2}}=\dfrac{{{q}^{2}}{{B}^{2}}r_{0}^{2}}{2m}$. (Since we substituted equation (1) in the kinetic energy equation).
Therefore the correct option is (c).

Note: There are two types of accelerators; linear accelerators (LINAC) and circular accelerators. Linear accelerator accelerates particles along a linear, or straight, beam direction. Circular accelerator accelerates particles around a circular path. In circular accelerators, dees are located between the poles of electromagnet which applies a static magnetic field B perpendicular to the electrode plane. The magnetic field causes the path of the particle to bend in a circle due to the Lorentz force perpendicular to their direction of motion. Cyclotrons cannot accelerate neutral particles.