> So, how come the entire board isn't melted? Just one little chip? 
> Presumably all that current had to get *in* to the board somehow...

Of course, but presumably the PCB tracks have low enough resistance not to 
instantly melt the PCB when a couple of amps goes through them.  The same 
can't be said for the tiny tiny wires inside an IC.  Remember, heating is 
due to power not current.

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