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How do I describe the energy an electron gains (and loses) in a circuit?

I am looking at the simple circumstance of an electron traveling through a long wire by a potential. I see how one would be motivated to use the usual formula, W=F*d, to describe the electron's net ...
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why is total electron energy of an electron in metal negative?

In my textbook, it says that any electron bound in metals, modelled as some potential well $U$, has negative total electron energy, as shown below in the figure. Why is the total electron energy ...