  In this paper we investigate a hybrid quantum system comprising a mechanical
oscillator coupled via magnetic induced electromotive force to an $LC$
resonator. We derive the Lagrangian and Hamiltonian for this system and find
that the interaction can be described by a charge-momentum coupling with a
strength that has a strong geometric dependence. We focus our study on a
mechanical resonator with a thin-film magnetic coating which interacts with a
nano-fabricated planar coil. We determine that the coupling rate between these
two systems can enter the strong, ultra-strong, and even deep-strong coupling
regimes with experimentally feasible parameters. This magnetomechanical
configuration allows for a range of applications including electro-mechanical
state transfer and weak-force sensing.
