The Comptroller, Treasurer, Corporation Counsel, Chief of Police, Police Judge, Street Commissioner, City Engineer and Building Inspector, respectively, before entering upon the duties of their respective offices shall execute a bond to the City in such penal sum as the City Council by ordinance may determine, conditioned for the faithful performance of their duties, including in the same bond the duties of all offices of which he is made by this Charter ex-officio incumbent. All official bonds shall be approved by the City Council, and when so approved, shall be filed with the Comptroller, except the bond of the Comptroller which shall be filed with the Mayor. All provisions of any law of this state relating to official bonds of officers shall apply to such bonds, except as herein otherwise provided. No City officer shall be eligible as a surety upon any bond running to the City as obligee. The City Council shall have power, whenever by it deemed expedient, to require a new or additional official bond of any officer.