An apparatus for inductively heating metal bodies, said apparatus comprising: - a handpiece arranged to be grasped by an operator, said handpiece including: - a hollow handle portion; - an elongated support extending from said hollow handle portion; - an electric inductor circuit comprising at least one inductor winding element that is arranged at an end portion of said elongated support; - a current transformer arranged in said hollow handle portion and electrically connected with said electric inductor circuit; - an electric generation unit electrically connected to said current transformer, and configured to supply an alternating current to said electric inductor circuit through said current transformer; - a hydraulic circuit of a cooling liquid, said hydraulic circuit comprising: - a cooling portion, defined within said handpiece, for cooling said hollow handle portion and said at least one inductor winding element; - a cooled portion having a volume to contain a predetermined amount of said cooling liquid; characterized in that said cooled portion of said hydraulic circuit comprises a plurality of parallel-arranged branches (6 1,6 2,...6 n) including respective reservoirs (60 1,60 2,...60 n), in that said volume is split into substantially equal parts among said reservoirs (60 1,60 2,...60 n), in that said parallel-arranged branches (6 1,6 2,...6 n) comprise respective temperature sensors (6 61,66 2,...66 n) configured to detect and notify respective temperature values (T1, T 2,... Tn) of said cooling liquid contained in said reservoirs (60 1,60 2,...60 n), and in that said cooled portion of said hydraulic circuit comprises a switching valve unit configured to selectively put each of said parallel-arranged branches (6 1,6 2,...6 n) of said cooled portion into hydraulic connection with said cooling portion of said hydraulic circuit, so as to selectively supply said cooling portion of said hydraulic circuit with said cooling liquid coming from a different reservoir (60 j) of said reservoirs when the temperature (T i) of said cooling liquid supplied by a branch (61 i, i ≠ j) of said parallel-arranged branches that is currently in hydraulic connection with said cooling portion exceeds a predetermined maximum temperature (T max). - a handpiece arranged to be grasped by an operator, said handpiece including: - a hollow handle portion; - an elongated support extending from said hollow handle portion; - an electric inductor circuit comprising at least one inductor winding element that is arranged at an end portion of said elongated support; - a current transformer arranged in said hollow handle portion and electrically connected with said electric inductor circuit; - a hollow handle portion; - an elongated support extending from said hollow handle portion; - an electric inductor circuit comprising at least one inductor winding element that is arranged at an end portion of said elongated support; - a current transformer arranged in said hollow handle portion and electrically connected with said electric inductor circuit; - an electric generation unit electrically connected to said current transformer, and configured to supply an alternating current to said electric inductor circuit through said current transformer; - a hydraulic circuit of a cooling liquid, said hydraulic circuit comprising: - a cooling portion, defined within said handpiece, for cooling said hollow handle portion and said at least one inductor winding element; - a cooled portion having a volume to contain a predetermined amount of said cooling liquid; - a cooling portion, defined within said handpiece, for cooling said hollow handle portion and said at least one inductor winding element; - a cooled portion having a volume to contain a predetermined amount of said cooling liquid; characterized in that said cooled portion of said hydraulic circuit comprises a plurality of parallel-arranged branches (6 1,6 2,...6 n) including respective reservoirs (60 1,60 2,...60 n), in that said volume is split into substantially equal parts among said reservoirs (60 1,60 2,...60 n), in that said parallel-arranged branches (6 1,6 2,...6 n) comprise respective temperature sensors (6 61,66 2,...66 n) configured to detect and notify respective temperature values (T1, T 2,... Tn) of said cooling liquid contained in said reservoirs (60 1,60 2,...60 n), and in that said cooled portion of said hydraulic circuit comprises a switching valve unit configured to selectively put each of said parallel-arranged branches (6 1,6 2,...6 n) of said cooled portion into hydraulic connection with said cooling portion of said hydraulic circuit, so as to selectively supply said cooling portion of said hydraulic circuit with said cooling liquid coming from a different reservoir (60 j) of said reservoirs when the temperature (T i) of said cooling liquid supplied by a branch (61 i, i ≠ j) of said parallel-arranged branches that is currently in hydraulic connection with said cooling portion exceeds a predetermined maximum temperature (T max).