Let's say you have a LED bulb that runs on a battery.

Let's say you have a LED bulb that runs on a battery. If you wanted to prolong the time this bulb runs on one charge, would you place a solar panel next to it, catch some of the light (Please remember, that light is not avalable for lighting up your room any longer, which is why you operated the bulb in the first place.), then use it to give the battery some additional charge?

No? Neither would I. First of all, I operate the bulb because I want light. Next, using a solar panel to reclaim some electricity will always have a bad efficiency, only very little of the electricity drained from the battery to run the bulb will return.

Now, would you do something like this for your phone? I mean, attaching something to your phone that will catch some of the RF power (which is then not available for contacting base stations any longer, so the phone needs to ramp up the power) and feed it back into the battery, would you do that? I know I wouldn't, but apparently there are people out there who still believe in perpetuum mobiles:

https://www.kickstarter.com/projects/nikolalabs/nikola-phone-case-power-your-phone-with-wasted-ene
https://www.kickstarter.com/projects/nikolalabs/nikola-phone-case-power-your-phone-with-wasted-ene