(a) Writing a mass dimension as $M$, a time dimension as $T$, a length dimension as $L$ and a charge dimension as $Q$, 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 $m$ and charge $q>0$ is accelerated by a constant electric field $\mathbf{E} \neq \mathbf{0}$. At time $t=0$, the particle is at rest at the origin.

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

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

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

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