MAGNETISM (III)

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We can calculate the Dipole Inductance. Let’s consider a cylindrical coil (14,3 m long and 9 cm wide on average) with 80 turns and a perpendicular magnetic field 8,33 T.
The magnetic Flux throughout the surface is: φ = N·B·S   ⇒  φ = 80·8,33·(14,3·0,09)   ⇒   φ ≈ 1000 Wb With,    φ = L·I L = 1000/11800   ⇒     L ≈ 0.1 H So the stored energy is: Ed = ½·L·I2   ⇒   Ed ≈ 7 MJ Considering 1232 dipoles: ET ≈ 9 GJ Enough to melt completely 45 tons of Gold from 25 ºC.
If we consider the dipole lenght, the energy density of the LHC magnets is: 7000/14,3 ≈ 500 kJ/m
In addition to just curving the beam, it is also necessary to focus it. Because protons are electrically charged a particle beam diverges if left on its own. Focussing the beam allows its width and height to be constrained so that it stays inside the vacuum chamber. This is achieved by quadrupole magnets, which act on the beam of charged particles exactly the same way as a lens would act on a beam of light
Let´s imagine that the positive particles (protons in LHC) come from the right. The first quadrupole takes control of the beam width while the second one does the same with the beam height. The two quadrupoles working together keep the protons tightly bunched so that the greatest number of collisions occur. There´s a total of 858 quadrupole magnets.
Other magnetic multipoles act to help in beam focussing and counteracting other interacctions that each beam suffers (gravitational interactions over protons, electromagnetic interactions among bunches, electron clouds from de pipe wall, etc). The next image shows the basic multipolar magnetic cell (FODO), 110 m long, in LHC.
There are, in addition, eight sets of so called "inner triplet" magnets in the LHC. Their job is to focus the particle beams into the four areas where particles will collide. The size of bunches passes fromm 0,2 mm to 16micrometers - at IP (ATLAS, CMS) 16 μm - in the triplets      ~1.6 mm - in the arcs          ~0.2 mm