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A tiny radioactive battery could keep your future phone running for 50 years
(www.techradar.com)
This is a most excellent place for technology news and articles.
The batteries are already connected to a "heatsink" in the phones we use now. Fast charging can be as lossy as 86% for lithium ion batteries qt very high charge rates, so the 60w fast charging will dissipate more than 6w of heat already.
And yes, the radiation battery either has to constantly use the power or it will be just pumping the voltage up until something starts to conduct