We study the quantum dynamics of the coupling between a cavity optical field and a cavity microwave field via the electro-optic effect. This coupling has the same form as the opto-mechanical coupling via radiation pressure, so all previously considered opto-mechanical effects can in principle be observed in electro-optic systems as well. In particular, we point out the possibilities of optical cooling of the microwave mode, entanglement between the optical mode and the microwave mode via electro-optic parametric amplification, and quantum-back-action-evading optical measurements of microwave quadratures.