Question

A cyclotron is used to produce a beam of high-energy deuterons that then collide with a target to produce radioactive isotopes for a medical procedure. Deuterons are nuclei of deuterium, an isotope of hydrogen, consisting of one neutron and one proton, with total mass 3.34×10−27kg. The deuterons exit the cyclotron with a kinetic energy of 5.00 MeV .

1. What is the speed of the deuterons when they exit? Express your answer with the appropriate units.

2. If the magnetic field inside the cyclotron is 1.25 T, what is the diameter of the deuterons' largest orbit, just before they exit? Express your answer with the appropriate units.

3. If the beam current is 350 μA how many deuterons strike the target each second? Express your answer as the number of the deuetrons.

Answer 1

1)

We have the kinetic energy of the deuteron when they exit,

(since,)

Given the mass of the deuteron,

Let the velocity of deuteron when they exit be,

So,kinetic energy of deuteron,

ie,equating kinetic energy,

So,

Velocity of the deuteron when exit,

2)

We have during the time of exit of the deuteron from the cyclotron,the maximum velocity of the cyclotron as,

The magnitude of the magnetic field applied,

Here,the magnetic force on the deuteron,

Also the magnetic force on the deuteron provides the neccessary centrepetal force for deuteron the continue in circular orbit,

ie,if the radius of the circular orbit,

So,centripetal force,

equating both gives,

So,raduis of the circualr path,

during the time of exit of the deuteron from the cyclotron,the maximum velocity of the cyclotron as,

mass of the deuteron,

Since the deuteron has only on proton and neutron,Charge of deuteron=charge of proton(Since the neutron is chargeless)

SO,during the time of exit,the maximum radius of the deuteron,

So,the maximum diameter of the circular path of deuteron when exits,

Diameter of largest orbit when it exits,

3)

Given the current of the deuteron beam as,

Let the total charge strikes the target in each seconds,

So,the Current of the deuteron or the charge per unit time(1 second),

ie,Here given the charge strikes in 1 second on target or current,

ie,

We have the charge of each deuteron,

So,the number of deuteron in charge Q,

The number of deuteron strikes the targetb each second,Dueterons