Question

How to find electric field intensity at a point?

Answer 1

The acceleration relies on synchronising the phase and frequency of the standing wave with the particle's transit through the cavity. At time zero the proton enters the resonator, and is accelerated forwards by the negative field gradient in the left half of the cavity. The frequency is adjusted so the period is twice the time the particle takes to cross the cavity, so after one quarter of the period, τ/4τ/4, the particle has reached the middle of the box and the field has fallen to zero. At this point the particle is coasting. As the particle passes the centre of the resonator the field changes phase by ππand the particle is again accelerated by the field gradient in the right half of the cavity.

In big accelerators like the LHC the proton velocity is effectively constant (at 0.99 and a bit cc) so the standing wave frequency is constant. It's just a matter of getting the phase right.