The Levitated Dipole Experiment (LDX) is an innovative facility to study plasma confinement in a dipole magnetic field, created by a superconducting solenoid (floating coil), which is magnetically levitated in the center of a 5 m diameter by 3 m tall vacuum chamber. This persistent mode, floating coil (F-coil) consists of a Nb3Sn magnet installed inside a high-pressure vessel filled with 12.5 MPa helium gas at room temperature. It is surrounded by a high heat capacity fiberglass-lead composite radiation shield and by a toroidal vacuum shell. The built-in tube heat exchanger serves to cool the magnet, the helium vessel, and the thermal shield. When positioned at the lower part of the vacuum chamber the F-coil is cooled by retractable cryogenic transfer lines to about 4.5 K and it is charged inductively by the charging coil installed outside of the vacuum chamber. Then the helium flow is interrupted, the heat exchanger is pumped out, retractable lines are disengaged from F-coil ports, the ports are plugged, and the F-coil is lifted to the middle of the chamber to initiate and study plasmas. After several hours and just before the F-coil warms up to about 10 K it is lowered down for the next re-cooling or for discharging. This paper describes the F-coil cooling system and inductive charging system operation and performance.