Zipping round and round

The Large Hadron Collider at CERN has a radius of \(R=4.5~km\) and uses \(B=8~T\) magnets to bend the particles around the accelerator. Estimate the energy of protons going around the accelerator in J?

The energy of a relativistic particle is calculated by \(E=\sqrt{p^2c^2+m^2c^4}\), with \(p\) is the particle's momentum, \(m\) is the particle's mass and \(c\) is the speed of light.

You may want to check this Minute Physics clip for a little background on relativistic momentum.

The proton's speed is extremely close to the speed of light.

Note: we're actually going to come out a touch high for the energy per proton, but it's still a pretty good estimate considering how simply we've modeled the LHC.

Details and assumptions

  • The proton charge is \(-e=|e|=1.6 \times 10^{-19}~C\) and the speed of light is \(c=3 \times 10^{8} ~m/s\). The proton's mass is not necessary in this problem.

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