Compare the efficiency of a colliding-beam accelerator with that of an accelerator in which the particles strike a target made up of identical stationary particles.

Answer

Hint . Compare the accelerating potentials needed to obtain the same internal energies of the bunch.

Explanation

The internal energy of a cluster in a conventional accelerator with a stationary target is `epsilon._0 =sqrt(2epsilon_0(2epsilon_0_evarphi))` (see Problem 13.6), in a colliding-beam accelerator it is `=epsilon_0.=2evarphi_c+2epsilon_0` . Equating both expressions, we obtain
`varphi=2varphi_c(2+evarphi_c//epsilon_0)`
where `varphi` is the accelerating potential in a conventional accelerator, and pc is the accelerating potential in a colliding-beam accelerator.
Clearly, for `evarphi_c ltlt epsilon_0` a colliding-beam accelerator is ineffective since the accelerating potential of a conventional accelerator is only four times greater than in the case of a colliding-beam accelerator. But in the ultra-relativistic case, when `epsilonvarphi_0 gt gt epsilon_0` the colliding-beam accelerator is very effective.