Path Independence

Work of Adway Dhillon (adhillon6)

Short Description of Topic

The Main Idea

This page is about an apparatus called the Cyclotron. This device is used to accelerate charged particles (like protons, neutrons, alpha particles, etc.) and ions to high energies. It uses a combination of strong magnetic and electric fields to accelerate the charged particles. The two fields (electric and magnetic) are perpendicular to each other, hence very aptly name "crossed fields". When the charged particle is accelerated by the cyclotron, the frequency of tis revolution in the two semi-secular plates (also called the "Ds") is independent of its energy. This is an important fact in the working of the device, as we will subsequently study in this lesson.

PRINCIPLE:
The working of the cyclotron is based on the fact that a positively charged particle can be accelerated to a sufficiently high energy with the help of smaller values of oscillating electric field by making it cross the same electric field time and again, with the help of a stringer magnetic field.

CONSTRUCTION:
A cyclotron device consists of two D-shaped hollow evacuated metal chambers (say D1 and D2) called the dees. These dees are placed horizontally with their diametric edges parallel and slightly separated from each other. The dees are connected to a high frequency oscillator which can produce a potential difference of the order of 10^4 volts at a frequency of about 10^7 hertz. The two dees are enclosed in an evacuated steel box and are well insulated from it. The box is placed inn a string magnetic field produced by two pole pieces of an electromagnets N,S. The magnetic field is perpendicular to the plane of the dees. At the centre of the device, P is a place of ionic source or positively charged (Check the diagram).

WORKING AND THEORY:
The positive ion to be accelerated is produced at P. Suppose, at that instant D1 is at negative potential and D2 is at positive potential. Therefore, the ion will be accelerated towards D1. On reaching inside D1, the ion will be in a field free space (as the two dees are hollow and metallic). Hence it moves with a constant speed in D1, say v. But due to a perpendicular magnetic field of strength B, the ion will describe a circular path of radius r (say) in D1.

Now in case the time taken by the positive ion to describe a semicircular path is equal to the time during which half cycle of the electric oscillator is completed, then as the ion arrives in the gap between the two dees, the polarity of the two dees is reversed, i.e. D1 becomes positive and D2 becomes negative. Then, the positive ion is accelerated towards D2 and it enters D2 with a greater speed, which remains constant in D2. The ion will describe a semicircular path of greater radius due to a perpendicular magnetic field and again will arrive in the gap between the two dees exactly at the instant, the polarity of the two dees is reversed. Thus, the positive ion will go on accelerating every time it comes into the gap between the dees and will go on describing a circular path of greater and greater radius with greater and greater speed and finally acquires a sufficiently high energy. The accelerated ops can be removed out of the dees from the window W, by applying at the elected field across the deflecting plates E and F (check the diagram).

A Mathematical Model

The charged particle describes a circular path of radius r (say) in D1, due to the perpendicular magnetic field B. Hence, the magnetic force experienced by the particle is balanced by the centripetal force on the particle due to its circular motion (in a circle of radius r).
This gives us the following expression:
[math]\displaystyle{ Bqv = mv^2/r }[/math]

Here: m and q are the mass and the charge of the ion respectively.

Therefore: [math]\displaystyle{ r = mv/Bq }[/math]

MAXIMUM ENERGY OF POSITIVE ION:
Let [math]\displaystyle{ v_0, r_o }[/math] = maximum velocity and maximum radius of the circular path followed by the positive ion in the cyclotron.

Then: [math]\displaystyle{ mv_0^2/r_0 = Bqv_0 or v_0 = Bqr_0/m }[/math]

Therefore: [math]\displaystyle{ Max K.E. = 1/2 mv_0^2 = 1/2 m(Bqr_0/m)^2 = B^2q^2r_0^2/2m }[/math]

CYCLOTRON FREQUENCY:
If T is the time period of oscillating electric field then: [math]\displaystyle{ T = 2t = 2pim/Bq }[/math]

The cyclotron frequency is given by

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

A Computational Model

Go to the following link to have a look at the picture: http://i64.tinypic.com/mil0sz.jpg

Examples

The Large Hadron Collider (LHC) is the world's largest and most powerful particle collider, and its functioning is based on the same principle as a simple cyclotron. It is the largest, most complex experimental facility ever built, and the largest single machine in the world. It was built by the European Organization for Nuclear Research (CERN). The LHC's aim is to allow physicists to test the predictions of different theories of particle physics, high-energy physics and in particular, to further test the properties of the Higgs Boson.

Connectedness

1. How is this topic connected to something that you are interested in?

I have been closely monitoring and following the developments at the Large Hadron Collider at CERN. The device's functioning and the underlying principle it works on got me interested in this field. This led me to study more on cyclotrons and particle accelerators.

1. How is it connected to your major?
2. Is there an interesting industrial application?

History

The cyclotron was first developed by Lawrence and Livingston at the University of California, Berkeley. Lawrence, having worked extensively on electromagnets before, was behind much of the theoretical work. On the other hand, Livingstone was responsible for translating that to a tangible device. When the device was created, it was the strongest particle accelerator of its kind. This was true till another accelerator was constructed in Leningrad, at the Radium Institute.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

[1] [2] [http://www.express.co.uk/news/world/565315/Scientists-at-Large-Hadron-Collider-hope-to-make-contact-with-PARALLEL-UNIVERSE-in-days/\

References

Baldwin, Paul. "Scientists at Large Hadron Collider Hope to Make Contact with PARALLEL UNIVERSE in Days." N.p., n.d. Web. 05 Dec. 2015.

"Latest Experiment at Large Hadron Collider Reports First Results." MIT News. N.p., n.d. Web. 05 Dec. 2015.

"Large Hadron Collider Starts Doing Science Again." Scientific American. N.p., n.d. Web. 05 Dec. 2015.