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It could be your connection to your battery. If its bumped loose, you would lose all power, even while driving.
Check the level of the oil in your engine. If it's ok, then you must have a faulty gauge.
Usually when the battery light comes on its the Alternator, or a bad wire from Alternator to battery, but id check Alternator first.
your Alternator
If it is your voltmeter that drops while the vehicle is running. Most likely the alternator isn't putting out enough volts.
The alternator is not working properly. Your car will still run off the battery for a little while then when the battery dies the car dies. Check the alternator or take it to autozone and they will check it for you.
the ground to the battery i had the same problem on my truck
Bad battery or alternator, or simply ran out of gas
That means you need to get toa gas station.
Yes, the wire gauge numbers are smaller. A typical automotive battery cable would be 4 gauge, the wires after the fuse box would be 10, 12, 14, 16, 18 and 20 gauge depending on the accessory.
If all the connections and wiring look good, it is probably the fuel gauge sending unit in the fuel tank.
If the alternator fails, it is no longer keeping the battery charged and your engine is running on nothing but battery power. Once the battery runs down, your engine will no longer run.