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I am much interested in physics. I have been caught up with one serious question related to speed of light. Hearing about Large Hadron Collider (LHC), it is said that protons in there are accelerated to almost the speed of light, i.e., $3\times10^8$ metre per second. One can be sure that the length in which the above experiment is conducted cannot exceed even $50~\text{km}$. (LHC at CERN is a $27$-kilometer ring buried deep below the countryside on the outskirts of Geneva, Switzerland.) So how can they achieve almost the speed of light in such a small place? Also, out of curiosity, I would like to know what instrument they use for measuring that speed?

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1) clarification: Lakh = 100,000. 2) area is not measured in km. 3) there is a problem with your keyboard: the shift key seems to play up when you type a title. Better to fix that. –  Johannes Aug 8 '13 at 3:09
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You might want to read this article that describes how the protons are accelerated.

The LHC uses eight 2MV RF cavities. Assume that the electron gets the full benefit of this (it doesn't - see below) so the energy of a circulating proton increases by 16MV with each circuit. The protons complete about 11,000 circuits per second, so their energy increases by about 0.18TeV per second and it would take about 40 seconds to accelerate to the full design energy of 7TeV.

It actually takes a lot longer than this (about 20 minutes according to the article I linked). The 2MV is the maximum field strength in the RF cavities, but the protons don't feel the full 2MV for their whole passage through the cavity.

A little while ago I answered a question about RF cavities, Accelerating electrons via microwaves, and you might find this interesting.

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They use extremely powerful magnets in order to accelerate the particles to enormous speeds. Because the particles have such small masses, it is possible to provide enough energy to get the particles to these speeds with current technology. In 2012, the LHC could operate at 4 TeV, which is enough energy to accelerate very light particles (such as protons) to immense speeds.

I am not 100% sure on how they measure speeds of particles, but I am guessing that there aren't instruments that could directly measure velocity. Instead, they use mathematics in order to determine how fast the particles are going. By knowing the energy put into a particle and its mass, it is possible to derive the velocity.

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The magnets have nothing to do with accelerating the protons or ions (which is done with microwaves), but rather with keeping the beam bending around in a circle so that the klystrons (the microwave cavities where the acceleration happens) can be used many times on each proton. Momentum is determined by the bending radius of the beam in the magnetic fields, and combined with know the mass of the beam particles gives velocity. Of course, velocity is a figure no one bothers to calculate except when writing a press release: particle physicists stick with momentum and energy whenever possible. –  dmckee Sep 8 '13 at 2:14
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