The particles they use have a net electric charge and moving electric charges create a magnetic field around them.
The accelerator is a gigantic loop of electromagnets, they turn on the electromagnets in a perfectly times series of pulses as the particle travels in circles around the loop. As the particle passes each magnet, the magnet’s field *pushes* the particles field speeding it up.
Around and around it goes, faster and faster and faster. etc.
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