If a computer is a distance away, on the boundary fringe or wifi or wireless Internet, does it have to “work harder” to load a browser or is it always “working” the same amount of work no more no less and just takes longer to load?
The wifi adapter on your computer will transmit at a higher power level so that the signal can get through as clearly as possible. There may be some dropped packets and retries and such, but the overhead from that is going to be fairly minimal. If it gets to the point where it’s no longer minimal, then your browser isn’t going to reliably connect any more and you’ll get all kinds of connection issues.
As far as your browser goes though, it’s probably drawing a tiny bit less power since it has to wait for the network to deliver the packets, and if the packets are being delayed due to retries then the browser will spend more time doing nothing, which doesn’t draw power.
So overall, your battery will die quicker due to the wifi and your processor will run (very) slightly cooler due to the decreased load.