= Solution
An RF cavity stores an oscillating <electromagnetic field>. Its longitudinal <electric field> does <work> on a correctly phased bunch, with energy gain $\Delta E=qV_{\rm RF}\mathcal T\sin\phi_s$ for a chosen phase convention and transit-time factor $\mathcal T$. The bunch arrival time is synchronized so successive gaps accelerate rather than decelerate it. The <magnetic force> guides and focuses the beam but does no work.
A <linear accelerator> uses new accelerating structures along its path; cell lengths and timing accommodate the particle speed, and each bunch normally traverses a given section once. A circular <synchrotron> reuses the same RF cavities each revolution. Its synchronous phase supplies both acceleration and radiation replacement; RF frequency or harmonic synchronization and bending-field strength follow the required orbit and particle speed. Longitudinal phase stability captures nearby particles into RF buckets.
For very high-energy \b[electron and positron] colliders, a linear geometry avoids repeated synchrotron losses: at fixed radius, loss per turn scales as $E^4/m^4$. This is not a statement that all high-energy colliders are preferably linear. The LHC was already a circular proton collider in the 2015 context; the proton's much larger mass greatly suppresses radiation at the same energy. Circular electron machines can also be competitive at lower energies. Magnet limits, luminosity, cost and beamstrahlung enter any real comparison.
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