What happens to a DC current at the end of a wire?
A way to look at your question is to look at the "circuits" you are talking about.
When a wire is not connected to another wire, i.e. exposed to the air, we call that "open circuit" and that means that no current is flowing through the wire. If there is no current, there is no DC current (I) and no power is wasted/transferred.
You can think about the air gap between to unconnected connectors as a "resistor" with a very high resistance (R).
From Ohm's Law:



We see why there is no current running in the "circuit" when the resistance is very large:



There is no power (or energy) transfer because electrical power (P) is:



When the current diminishes so does the power:



When you connect the connector, there is negligible resistance between the two conductors (wires) and the resistance of the circuit will be set by the device you are connecting to it (the Engineers designed the device to have the correct resistance for the application).

IS that helpful at all?

*Source: I'm an engineer.