If I had sense, I'd point you to a web site. Instead, I'll try myself...
DC Circuits are pretty simple. A battery is a DC, or direct current, power source. One terminal is positive, one is negative. Current flows in one direction. This is in contrast to AC current, as from a household outlet. In AC, the polarity of the wires switches back and forth, or alternates, as does the current. The big difference is that there's a large magnetic field generated with current changes direction, which allows transformers to work, and motors to be efficient. (Transformers don't work at all with DC.) Batteries are chemical devices, so naturally don't make AC. I'll ramble on about DC, and drop AC...
With DC, you can look at electricity like water flowing through a pipe. The pressure in the pipe is the voltage, the amount of water flowing through the pipe is the current, the resistance to the flow of water is resistance. Now the trick is with electricity, we don't have the electricity flowing all over the place. It goes in a loop. Disconnect either terminal from a battery, and no current flows. With voltage, it's the voltage between two points that causes current to flow. More voltage, more current. The lower the resistance (bigger wire, bigger bulb, etc.), the more current. Current (in amps) = Voltage (in volts) divided by Resistance (in ohms). You'll sometimes see this as i=e/r, where i is the symbol for current, e for electromotive force or voltage, and r resistance. This is Ohm's law. If you know any one two of the three, voltage, current, resistance, you can find the other.
The other formula is p=i*e, or power (in watts) = voltage multiplied by current. Power is, well, a unit of power. The amount of work done. In a light bulb, the current x voltage gives the total output of heat and light from the bulb.
These two formulas are why for a light bulb, you'll typically see a voltage and current rating, like 12.0 Volt, 2 amp; or a voltage and power rating, like 12.0 Volt, 24 watt; but not all three numbers. If you know two, you can find the third.
Batteries are typically rated by voltage (for a normal AA cell, it's 1.5V), and capacity, which is how long they can supply an amount of current before they expire. Capacity is usually in mAH, or mili-amp hour, which is how many 1000's of an Amp they can put out in one hour. (But usually rated at 10 hours or so. So a 1300mAh battery would be capable of producing 130 miliamps (.13 amps) for 10 hours.)
Now, this would all be really simple, except for a couple of things:
1. The resistance of an incandescent bulb changes a LOT with the voltage. They have a very low resistance when off, and as you apply current the resistance goes up as the bulb gets warmer. So applying twice the voltage to a bulb won't result in twice the current. The equations are still accurate, it's just you can only use them at one voltage range.
2. The resistance of an LED is even more complicated. They allow almost no current to flow until you get to a certain voltage, then the current goes up *very* quickly with voltage. They're called a non-linear device, which incandescent bulbs really are too, just LED's are more so. So with an LED you have to do one of two things: Either drive the LED at a voltage below that which will blow up the LED (either from a battery or set of batteries with known voltage), or limit the current somehow with a resistor or electronic regulator.
3. Batteries don't put out a constant voltage throughout their life, it drops. (For some batteries it goes up a little bit at first, as the battery gets warm.) For alkaline and carbon-zinc batteries, the voltage mostly gradually goes down. This is why you'll see a light get dimmer with use. For ni-cad and ni-mh batteries, the voltage drops much less during use, until the end, where it screams down very quickly. This is why rechargeable lights (and phones) die FAST. Lithium cells also have a gradual voltage drop.
4. Batteries have an internal resistance. That is, if you connect a huge wire from the positive to negative terminal, you won't get infinite amps, you'll get way less. Given the same type of chemistry (alkaline, ni-cad, etc.) larger batteries have lower internal resistances. They put out closer to full voltage with the same load. Old carbon-zinc and 'heavy duty' batteries have a high internal resistance. Alkalines are much lower. Ni-cads, Ni-Mh's, and lithiums are much lower still. With some Ni-Mh's, and all but coin cell sized lithiums, the internal resistance is so low that shorting the battery could result in so much current flowing, and thus heating of the battery, that it could go "boom". Lithium 123 cells have a thermal shutdown gizmo built into them to keep this happening.
Ok, so because of all this non-linear stuff, the only real practical way to tell how long a flashlight will be so bright is to test it. You'll see 'run time plots' (thanks Roy!) charting the output of a flashlight over time, with a given battery. Change the bulb, or the battery, and the graph changes. This is one of the fun things of playing around with flashlights.
Oh, and the light output of a lightbulb isn't directly proportional to the current, either, but then by now you probably didn't expect it to be....
Ok, done rambling, hopefully everyone all confused now.
(p.s. The mAH rating of a battery isn't linear, either. A battery has a higher capacity with lower loads than higher ones. So a 1300mAh battery may last 150 hours at a load of 13 mA, instead of 100, but only 45 minutes at a load of 1300mA. With rechargable batteries it's not too bad, but with disposable cells it can be. Fun?
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