Whenever the county board shall determine after hearing that any county officer or deputy is physically or mentally incapable of performing the duties of his office and cannot recover sufficiently to be able to perform such duties, the board shall declare such position vacant and appoint a successor. The county board shall appoint two physicians to examine the officer or deputy and shall receive the report of the physicians as evidence at the hearing. If the person so removed is an officer, the appointment shall be for the unexpired portion of the term. Such hearing shall be held only after ten days' notice in writing to the officer or deputy concerned. An appeal to district court may be taken from the action in the same manner as appeals are now taken from the action of the county board in the allowance or disallowance of claims against the county. The provisions of this section shall not apply to the office of county judge.