How big a magnetic field does the LHC require to contain its protons?

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In the LHC protons are accelerated to almost the speed of light. They are contained in the circular accelerator by the 1232 superconducting dipole magnets. How strong does the magentic field need to be?

If the LHC is at its maximum design energy, the protons have an energy of 7TeV. One eV (electron volt) is the energy gained by moving one unit of charge (protons and electrons have one charge unit q) through a potential of 1 Volt.

 

From AS level electromagnetism we know that the force on a charged particle with velocity v in a magnetic field B is

F = q v x B

and the magnitude of the force required to keep an object of mass m in a circlular path with velocity v is

F = m v2 / r

Equating the two (assuming the velocity and magnetic field are perpendicular) and multiplying both sides by m give

m2 v2 / r = q m v B

so

mv / q r = B 

In relativity  E2 = p2 c2 - m2 , but the particle is travelling super relativistically so p2 >> m2 so E is approximately pc (and p = mv) in this case so

E / q r c = B

The energy E in Joules is q x 7TeV. There are 1232 magnets each of length 15m

B = 7 x 1012 x 2 x pi / 1232 x 15 x c

B =  8T

Mark S. A Level Maths tutor, GCSE Maths tutor, A Level Further Mathem...

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