Why do electrons flow through a wire?
Its the voltage. If you measure the potential along a wire that connects a + and - terminal, there will be a gradual voltage drop all along the wire. Like, if you connect a 1.5V battery to long wire (that say has 10 Ohms of resistance because it is long and skinny wire), then you will get 150mA of current flowing through the wire. The "current" flows from the + terminal to the - terminal, and the actual physical electrons are flowing in the opposite direction.
In very simple terms: As long as there is a difference of potential between the ends of the wire (and the wire is not broken or somehow have infinite resistance), the negatively charged electrons from the negative side of the source voltage are effectively pushed to the positive end of the wire (like charges repel, unlike charges attract). This current flow (+ to - or - to positive...depending on your convention) will continue until the difference in potential becomes 0 (dead battery, open circuit, etc).
So those explanations are nice, here's an even better way to understand it in my opinion: Ok, so what are electrons? Negatively charged particles called electrons. What is the wire made out of? Metal, which is positively charged particles called protons. One last thing, what's a battery? It's something that's positively charged at one end and negatively charged at the other. So let's look at a picture of this! |dw:1345084374871:dw| Alright, so basically all that happens is the wire (made of protons, remember?) don't move, but the electrons hanging around the protons do. Negative and positive charges attract, right? So the electron nearest to the positive terminal of the battery is attracted to it and jumps over. So in a picture of the wire (Made of cute little positively charged protons, you didn't forget, no lol?) the electron jumped towards the positive terminal of the battery leaving a little hole there. Without an electron there to neutralize the charge, it causes that part of the wire to be positive and what would you know? It causes that next electron down the line to attract to it leaving another hole. |dw:1345084502761:dw| And thus, you have electron flow, electricity. Of course there's a little more going on in the battery chemically speaking, otherwise the battery would neutralize almost instantly. But that's another discussion for another day... lol
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