I am not sure that I buy that. Charging the battery moves molecules from one electrode to the other. The number of atoms in the battery doesn't change. Do they really get heavier by virtue of changing their arrangement?
If you use that logic (and maybe it is right) then it seems to me as though pushing a rock up a hill makes it ever so slightly heavier too.
Maybe that's my complete lack of intuition about non-classical stuff though.
Yes, they do get heavier by virtue of changing their arrangement. And yes, pushing a rock up a hill makes it - well, the rock will get slightly lighter in weight by virtue of being further from the Earth (classical inverse square law of gravity) - but the total mass of the rock-Earth system (assume you used solar power so you didn't take the energy from a terrestrial source) will slightly increase.
This is my understanding: say you have a spring which is just two positive particles. When you compress it, bringing the particles closer, you're storing energy in the EM field (E²+B²) near the particles. When you accelerate the spring, you're also accelerating the energy, which requires extra energy. This feels the same as the spring becoming heavier.
This is just a microscopic explanation for the general principle that energy has mass. I've never done the math to confirm that the extra energy to accelerate the compressed spring corresponds to the extra mass from stored energy.
When oxygen combines with hydrogen to form water, it gives off energy equivalent to the change in mass (water molecules are oh-so-slightly less massive than their constitute oxygen and hydrogen atoms). A battery produces power by converting molecules into different molecules with more binding energy (i.e. lighter molecules that have given up more energy/mass).
If you use that logic (and maybe it is right) then it seems to me as though pushing a rock up a hill makes it ever so slightly heavier too.
Maybe that's my complete lack of intuition about non-classical stuff though.