# Distance travelled in a Cyclotron

• Uku
In summary, the conversation is about finding the distance traveled by a deuteron in a cyclotron, given known data such as final velocity, frequency, magnetic field, radius, accelerating voltage, and charge and mass of the deuteron. The attempted solution involves calculating the gained velocity per turn, the number of turns in the cyclotron, the time spent, and the acceleration. However, a simpler solution is suggested by finding the change in kinetic energy and using the final kinetic energy and number of cycles to calculate the total distance traveled, resulting in a value of around 244m.
Uku

## Homework Statement

Known data:
$$v_{final}=3.99*10^{7}m/s$$
$$f_{osc}=12MHz$$
$$B=1.6T$$
$$R=0.53m$$
$$V_{acc}=80kV$$
$$q_{deut}=1.602*10^{-19}C$$
$$m_{deut}=3.344*10^{-27}kg$$
answer according to Halliday and Resnick x=240m

I got it all, and I have to find how much distance was traveled by the deuteron in the cyclotron, from starting of acceleration to the end of it.

## The Attempt at a Solution

From the potential across the dees I get

$$v_{gained_per_turn}=\sqrt{\frac{2E}{m_{deut}}}=2768586.602m/s$$

Since I know the final speed I can find the amount of turns the deuteron was in the cyclotron

$$n=\frac{v_{final}}{v_{gained_per_turn}}=14.41turns$$

Since f is constant, so is T and I can find the amount of time spent in the cyclotron:

$$t_{spent}=n*T=n*\frac{1}{f_{osc}}=1.2*10^{-6}s$$

The acceleration must be constant, the accelerating voltage is not changing

$$a=\frac{v_{gained_per_turn}}{T}=3.32*10^{13}\frac{m}{s^{2}}$$

Now

$$x=\frac{1}{2}a*t^{2}=23.92m$$

The answer in the book is 240m, which differs from mine ten times, smells like a power error?
Or am I doing something wrong?

you don't need to find the time and stuff..
Change in Kinetic energy, ΔK=2qΔV (ΔV is the difference in potential and factor 2 because the particle is accelerated twice in one cycle)
Final Kinetic energy is Kf=(qBR)2/2m (use B=2πfoscm)
then no. of cycles, n=Kf/ΔK
Total distance covered = n×2πRavg (where Ravg = R/√2)
You'll get around 244m

## 1. How is distance travelled calculated in a cyclotron?

Distance travelled in a cyclotron is calculated by multiplying the speed of the particle by the time it takes to complete one revolution in the magnetic field. This distance is known as the circumference of the cyclotron and is an important factor in determining the efficiency of the cyclotron.

## 2. What factors affect the distance travelled in a cyclotron?

There are several factors that can affect the distance travelled in a cyclotron, including the strength of the magnetic field, the energy of the particles, and the size of the cyclotron itself. Higher magnetic fields and particle energies will result in longer distances travelled, while a larger cyclotron will have a larger circumference and therefore a longer distance travelled.

## 3. How does the distance travelled in a cyclotron impact the efficiency of the machine?

The distance travelled in a cyclotron is directly related to its efficiency. The longer the distance travelled by the particles, the more times they will pass through the accelerating electric fields, resulting in a higher probability of interaction and therefore a higher efficiency. A longer distance travelled also allows for a greater range of particle energies to be achieved, making the cyclotron more versatile.

## 4. Can the distance travelled in a cyclotron be controlled?

Yes, the distance travelled in a cyclotron can be controlled by adjusting the strength of the magnetic field and the energy of the particles. By changing these factors, the circumference of the cyclotron can be altered, resulting in a shorter or longer distance travelled by the particles.

## 5. How is the distance travelled in a cyclotron measured?

The distance travelled in a cyclotron can be measured using a device called a Faraday cup. This cup collects the particles that have travelled a certain distance and measures their energy, allowing for the calculation of the distance travelled. Other methods, such as using a beam stop, can also be used to measure the distance travelled in a cyclotron.

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