Energy loss at the LHC

The Large Hadron Collider is the most advanced particle accelerator in the world, capable of accelerating protons to a total energy of $$1.12~\mbox{micro-Joules}$$ per proton. It's also really, really big: it's in the shape of a circle $$27~\mbox{km}$$ in circumference. Why, you may ask, does it have to be so big? After all it'd be cool to build one in your basement. The LHC is a circle so that the protons can be accelerated by the same process and equipment many times as they go around and around. Since the curving of the path is done by magnets, the large radius is necessary because we can only make magnets that are so strong.

There is another way that the large radius helps. If protons, which are charged particles, are moving in a circle then they have centripetal acceleration. Now, when charged particles accelerate they emit electromagnetic radiation. This is how radio antennas work. Electrons are accelerated and form a current in the antenna that is changing with time, producing radio waves. Radio stations take power to run, and radio stations in the United States, for example, put $$10^4$$ to $$10^5$$ Watts of power into their radio signals.

Similarly, the accelerating protons in the Large Hadron Collider emit electromagnetic radiation, which we call synchrotron radiation, and lose energy. This tends to slow them down. The power radiated away in synchrotron radiation by a single proton traveling in a circle of radius r is given by

$$P=\frac{2Ke^2p^4}{3m^4c^3r^2}$$

Note that the power loss goes down the larger the radius of the circle. This is an additional benefit of big accelerators: we lower the energy we need to put into the particles just to keep them moving in a circle. As an example, one bunch of protons (out of many) that zips around the LHC contains around $$1.15 \times 10^{11}$$ protons. How much power in Watts does the LHC need to provide to keep this bunch of protons moving at a constant speed? You may want to check this Minute Physics clip for a little background on relativistic momentum.

Hint: as you will see, synchrotron losses aren't a huge issue for the LHC, although for a smaller accelerator with electrons (which are much lighter than protons) synchrotron losses are significant.

Details and assumptions

• $$K=9 \times 10^9~\mbox{N}~\mbox{m}^2/\mbox{C}^2$$ is the electrostatic constant.
• $$e=1.6 \times 10^{-19}~\mbox{C}$$ is the charge of the proton.
• $$m=1.67 \times 10^{-27}~\mbox{kg}$$ is the mass of the proton.
• $$p$$ is the momentum of the proton.
• $$c$$ is the speed of light.
• $$r$$ is the radius of the circle.
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