If your experience level is comfortable working with aperture and shutter speed, and if you want to be able to manage your flash picutres, then this is what you need to know. These first basics are the dividing line where we become knowledgeable, no longer clueless. We won't be able to go very far without these flash fundamentals.
These basics are about "light itself", and are obviously important even if you always use automatic TTL metered flash.
Some people avoid any math. This one first page has a little simple multiplication in it, grade school stuff. It will not do you any harm by simply reading it. And relax, the following pages are better, but the first concern might be "how bright is my flash?"
1. Inverse Square Law
Light intensity falls off rapidly with distance from its source. This is called the Inverse Square Law, which says the intensity varies with the square of the flash-to-subject distance, this way:
2x the distance is 1/4 as bright, and 1/2 the distance is 4x brighter (2 stops)
3x the distance is 1/9 as bright, and 1/3 the distance is 9x brighter (8x is 3 stops)
4x the distance is 1/16 as bright, and 1/4 the distance is 16x brighter (4 stops)
etc.
Inverse Square Law is just a fancy name for a rather simple concept. Think of a flashlight - as the beam travels farther away from the source, the beam spreads out to illuminate a larger area, but becoming more dim with distance. All light spreads this way, from your flashlight, your photo flash, your table lamp, and your corner street light spreads this way too. All we are saying is that the light covers a larger area as it travels and spreads out. The greater area results in less light intensity per unit of area, simply because the light is the same power distributed over a larger increasing area. Here is that same Inverse Square Law link again, for their picture. See? The simple math shown there shows pictorially why, when any angle travels twice as far, its Width and Height spread to be twice as great. That 2W x 2H makes 4x Area which still contains the same light, which is therefore diluted to be only 1/4 as strong at 2x distance. 4 is the square of 2. It is really that simple, only about the spread of any angle in space, and not really about light at all, except that light spreads out with distance, and becomes weaker intensity. You know this, the flashlight becomes dim with distance, and will only reach so far. So the point is, light falls off fast with distance, inversely with the square of the distance. Your flash is a light, and it does this too.
Sunshine works exactly the same way too. However it may seem not to, simply because we are 93 million miles from the sun, and another few miles to yonder mountain we see here on Earth is a totally insignificant difference. Even the 240,000 miles to the moon is insignificant (1/4 of 1%), so the astronauts could use the same Sunny 16 rule there that we use here. On Mars however (half again farther from the sun than the Earth), they will have open up one more stop. But sunlight does appear constant to us here on Earth. From our viewpoint on Earth, sunlight uniquely does not vary with subject distance, which may give photographers false notions about how light actually works. But the flash is in the same room with us, only a few feet from the subject, so we WILL see the Inverse Square Law in action. It is a major factor.
Since intensity varies with distance from the light source, another implication is that any flash exposure can only be "correct" at one distance from the light source. That is a good thing to know. This Inverse Square Law (light falloff with the square of the distance) is true of all light, any light, a table lamp or a campfire at night, etc, but using flash for photos is commonly where this becomes more important to us to know. We cannot "fix" this Inverse Square Law situation, nor can we ignore it - we can only learn to work with it.
The exposure does not depend on where the camera is, or how far it is from the subject (unless the flash is on the camera). What matters is how far the flash is from the subject.
Exposure variation due to flash-to-subject distance
For reference, we know that one stop of exposure is a 2x brightness difference, and two stops is 4x.
Suppose the subject is at 8 feet from the flash, and everything is setup to be correctly exposed there. We can be certain that background objects at 11 feet will be underexposed 1 stop, and objects at 16 feet will be underexposed 2 stops. Foreground objects at 5.6 feet will be one stop overexposed, and objects at 4 feet will be 2 stops overexposed. You probably recognize those example distance numbers (4, 5.6, 8, 11, 16) as being f/stop numbers, and coincidentally (due to both definitions using squares), this aperture scale we have memorized provides a good quicky guide to estimate this falloff. If the subject distance approximates any f/stop number, then half or double power corresponds at one stop invervals in the same way. Merely coincidentally true, but it works, a reasonable guide. Technically, the square root of 2, or 1.414 times the distance is one stop down, and 2 times the distance is 1/4, or 2 stops down (which coincidentally agrees with the f/stop numbers if they are used as a distance).
The Inverse Square law explains why the room is seen being darker behind nearby people in a snapshot using direct flash. The distant background obviously has to be darker, it is farther from the flash (just how life is). However... we can often bounce the flash onto the ceiling, where it goes additional distance up to the ceiling and back down, and then the rest of the room simply lights up (if the room is not too large). The difference in subject distances is simply a smaller percentage of the longer bounce light path (which also requires substantially more flash power). We said before that flash exposure could only be correct at one distance from the source, but the bounce light travels up to ceiling and back down, and most places in the room are in fact at about that same disance from the ceiling, measured that way... so the entire (small) room is nearly equally exposed, more or less. Not without limits, but try it, it really works. Bounce flash has other good properties too (very soft light, Part 3 here), and is a wonderful effect at most opportunities with hot shoe flash. For a hot shoe flash, bounce should be the standard routine case whenever possible .
Note that for any Inverse Square Law distance computation (where half the distance is expected to give 2 stops more intensity), that distance is measured from the light source, which is from the flash tube to subject. For a softbox or shoot-through umbrella, the distance is from the flash tube through the fabric to the subject (distance of subject to light stand pole is close). For a reflected umbrella, this is the distance from the flash tube to the umbrella fabric, and then back to the subject (includes two trips along the umbrella shaft). The fabric is not the source of light, it merely looks that way.
Guide Number
If you meter your flash, either via TTL flash automation, or by using a hand held flash meter, then you don't really need this part, but it is still a prime fundamental, and will still be good to know.
Camera speedlights normally provide specifications for Guide Number (GN) as a guide to the flash power and its distance capability. Specifically GN is a tool for manual flash power levels with direct flash to automatically and invisibly deal with the Inverse Square Law. For any given "same flash-power" situation, we need to know that Guide Number is a constant, numerically equal to the aperture number (like the number 8 in f/8) multiplied by the subject distance. This constant GN (in this same flash power situation) is initially determined by the trial situation seen to give correct exposure. Or we can use the manufacturers chart of Guide Number (trial is what they did).
Guide Number = Aperture x Distance.
Aperture = Guide Number / Distance
Distance = Guide Number / Aperture
If for example, if f/8 is seen to give the correct exposure at 10 feet, then the Guide Number for this situation is determined to be 80 (from f8x10 feet). Or this Guide Number chart is in the flash manual. The advantage of knowing this Guide Number constant is that if we then move the light to be 5 feet from subject, then it tells us that GN80/5feet = f/16 will give us correct exposure there too. Or if we open the aperture to f/4, then the correct distance for this flash power will be GN 80/f4 = 20 feet. This Guide Number 80 is a constant (in this same flash power sitution), and its purpose is to make the inverse square law be trivial to compute.
You can work in either feet or meters. Since there are 3.28 feet in one meter, the GN in feet is simply 3.28 times the GN in meters. The Guide Number chart normally gives both values, the two usually shown as meters/feet.
Due to coincidence (aperture numbers increase by the square root of 2 to give half intensity, and the Inverse Square Law distance decreases by the square root of two to give double intensity), these square factors offset in the math, so that the simple product (aperture x distance) is a CONSTANT for correct exposure (for this given flash situation). Don't worry about the math in the definition of this Guide Number constant. Enough to know that the Guide Number automatically accounts for the Inverse Square Law, making it be almost trivial for us - as follows:
From knowing this Guide Number constant (GN = aperture x distance) for one flash situation (power and spread angle), we can recompute any other aperture/distance combination for correct exposure, which automatically takes the inverse square law into account. For example, if we know GN is 80 (feet), then we immediately know that f/8 at 10 feet, or f/10 at 8 feet, or f/4 at 20 feet or f/20 at 4 feet, will be just a few of the combinations giving the correct exposure at GN 80. This is a lot to know, and it really could not be any easier.
Note that the Guide Number charts are typically always printed showing ISO 100 values, but we can just multiply ISO 100 GN values by 1.4 to get ISO 200 GN values, or by 2 to get ISO 400 GN values, or by 2.8 to get ISO 800 GN values, or by 4 to get ISO 1600 GN values. Or we divide if converting going the other way.
Example of use: Suppose we plan to use f/8 at 12 feet at ISO 400. So we know we need flash power of f8 x 12 feet = GN 96 (feet) at ISO 400. Converting this to ISO 100 (to match the manuals chart) is GN 96/2 = GN 48 (feet, ISO 100). We search in the speedlight manual's guide number chart, and maybe we find the value at 24mm zoom and 1/4 flash power to be say 49. More than close enough to 48. This says that 24mm flash head zoom and 1/4 flash power is a close approximation of the power this f/8 12 foot setup needs. Duplicate this situation (set flash to 24mm zoom, 1/4 power, 12 feet) and set the camera at ISO 400 and f/8, and you're very close on first try.
Maybe more than you want to know, but if you are going to use or compare Guide Numbers, then you have to know this tricky part (many errors are made by not understanding this part). This Guide Number is only valid for the same flash power situation, specifically, for the same reflector and angle of coverage of the same flash power. Distribute the same "power" over more area and the "intensity" goes down. Concentrate the same power into a more narrow beam, and the intensity goes up. (The "power per unit area" times the coverage "area" is the same flash power level). If the guide number of one flash is specified as GN 180, but its zoom coverage angle is only illuminating a small spot on the wall, and another flash specifies GN 120, but it is illuminating the entire wall, then the way to bet about the most "power" is on this second unit. Angular coverage requires a lot of power, maybe more than a small flash has. GN is the illumination inside that small area, but we can only compare "power" for similar coverages. Make the coverages equal in size, and then compare the guide numbers. Part of something else, but this point is discussed more Here.
So - speedlight manuals generally have a chart of many different Guide Numbers, a different GN for each zoom setting (angle of coverage), and for each power level. A larger GN if the flash head is zoomed in tight, or a smaller GN if zoomed out wide. There is a Guide Number chart in the flash manual. This published GN is always speaking of unmodified direct flash, and usually for ISO 100 by convention. Any different ISO, or different power level setting, or zoom angle, or bounce or umbrella or any other modifier will change it to be a new situation, when any old guide number is no longer applicable.
In contrast, studio lights use a large variety of unknown light modifiers (umbrellas, softboxes, grids, snoots, etc), and so do not bother with Guide Number, since every case would be very different. Those studio photographers simply use handheld flash meters to measure what is actually happening. The studio light manufacturers instead publish an input power level in watt seconds (or may be called joules in Europe, 1 joule is equal to 1 watt second), which is a maximum capability.
Math: Doubling watt seconds of power is one stop exposure increase, and 4x power is 2 stops. Said another way, setting the flash to 1/2 power is one stop down, and 1/4 power is two stops down. Same as double ISO is 1 stop and 4x is 2 stops. But Guide number is mathematically different, multiplying Guide Number by 1.414 is one stop, and Guide Number x 2 is 2 stops (aperture number is one factor of Guide Number, and aperture number works that same way).
An example, suppose some flash situation (say ISO 200, full power, zoomed to 50mm) gives correct exposure at 25 feet at f/8. By definition, then we have determined that this guide number is 25 x 8 = GN 200 (feet) at ISO 200. The ISO 100 Guide Number chart for 50mm shows GN 144 (feet) at full power. So to compare our result to the ISO 100 chart, then GN 200 / 1.414 = 141 at ISO 100. Or conversely, the ISO 100 chart showing GN 144 tells us we can expect GN 144 x 1.414 = GN 203 at ISO 200.
