Question

Maximum kinetic energy of the positive ion in the cyclotron is

Answer 1

Step 1: Cyclotron

1. A cyclotron is a type of particle accelerator.
2. It accelerates a particle of charge $q$ and mass $m$ that enters with a velocity $\overrightarrow{v}$ into a region of a uniform magnetic field $\overrightarrow{B}$.
3. When a charged particle enters a uniform magnetic field normally it describes a circular path of radius $r$.
4. Otherwise, the particle moves parallel $\overrightarrow{B}$ with constant speed and at the same time gyrates about $\overrightarrow{B}$.
5. Therefore, the particle moves in a helix with $\overrightarrow{B}$ as axis.
6. This property is also used in devices called magnetic lenses for focusing beams of charged particles.

Step 2: Maximum kinetic energy of positive ion

The particle energy inside a cyclotron is given as

$E=\frac{q^2{B}^2{r}^2}{2m}$

Let us assume $r\text{'}$ to be the maximum radius of the circular path of the positive ion.
Let us assume $v\text{'}$ to be the maximum speed of the positive ion.

Upon equating the magnetic force to the necessary centripetal force we get

$qv\text{'}B=\frac{mv{\text{'}}^2}{r\text{'}}$

$\Rightarrow v\text{'}=\frac{qBr\text{'}}{m}$

Therefore, the maximum kinetic energy of the ion

$KE=\frac{1}{2}mv{\text{'}}^2$

$=\frac{1}{2}m{\left(\frac{qBr\text{'}}{m}\right)}^2$

$=\frac{B^2{q}^{2}r{\text{'}}^2}{2m}$

Hence, the maximum kinetic energy of the positive ion in the cyclotron is $\frac{B^2{q}^{2}r{\text{'}}^2}{2m}$.