a Cosmic Zevatron
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Accelerating a charged particle to ZeV energies is not easy! We can express how difficult it is in a way that is almost independent of the detailed physics of the acceleration process. I will illustrate this point with several lines of reasoning that lead to a common relationship first emphasized by Hillas (1984).

The simplest constraint would be that the accelerated particles must be contained inside the accelerator while they are being accelerated. So, we must demand that the particle gyro radius be less than the size of the accelerator. That is $$r_g = \frac{pc}{ZeB} = \frac{E}{ZeB} < R,$$ where $E$ is the particle energy, or $$E < ZeBR = 0.9 Z B_G R_{pc} Z~~{\rm ZeV}$$ where $R$ is a characteristic size for the accelerator.

Magnetic fields cannot accelerate charged particles; only electric fields can do that. It is not always easy to express the physics of the accelerator in terms of an applied electric field, but if it is, then obviously the maximum energy that can be achieved is given by $E < Ze{\sf E}R$, where [E]{} is the electric field. For a unipolar inductor, for example, ${\sf E} \sim v_{acc} B/c = \beta_{acc} B$, 