Paper 4, Section II, A

Dynamics and Relativity | Part IA, 2019

(a) Writing a mass dimension as MM, a time dimension as TT, a length dimension as LL and a charge dimension as QQ, write, using relations that you know, the dimensions of:

(i) force

(ii) electric field

(b) In the Large Hadron Collider at CERN, a proton of rest mass mm and charge q>0q>0 is accelerated by a constant electric field E0\mathbf{E} \neq \mathbf{0}. At time t=0t=0, the particle is at rest at the origin.

Writing the proton's position as x(t)\mathbf{x}(t) and including relativistic effects, calculate x˙(t)\dot{\mathbf{x}}(t). Use your answers to part (a) to check that the dimensions in your expression are correct.

Sketch a graph of x˙(t)|\dot{\mathbf{x}}(t)| versus tt, commenting on the tt \rightarrow \infty limit.

Calculate x(t)|\mathbf{x}(t)| as an explicit function of tt and find the non-relativistic limit at small times tt. What kind of motion is this?

(c) At a later time t0t_{0}, an observer in the laboratory frame sees a cosmic microwave photon of energy EγE_{\gamma} hit the accelerated proton, leaving only a Δ+\Delta^{+}particle of mass mΔm_{\Delta} in the final state. In its rest frame, the Δ+\Delta^{+}takes a time tΔt_{\Delta} to decay. How long does it take to decay in the laboratory frame as a function of q,E,t0,m,Eγ,mΔ,tΔq, \mathbf{E}, t_{0}, m, E_{\gamma}, m_{\Delta}, t_{\Delta} and cc, the speed of light in a vacuum?

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