Jurisdiction, Power Authority?

A county police officer has jurisdiction, therefore authority, in his county where he is recognized. If he leaves his county, his jurisdiction, he has no authority, and therefore no power. Not until he comes back into his jurisdiction will his power be recognized.

Jurisdiction from Wikipedia

Jurisdiction (from the Latin ius, iuris meaning "law" and dicere meaning "to speak") is the practical authority granted to a formally constituted legal body or to a political leader to deal with and make pronouncements on legal matters and, by implication, to administer justice within a defined area of responsibility. The term is also used to denote the geographical area or subject-matter to which such authority applies.