Effective Guide Number

The inverse-square law states that the intensity of a point light source is inversely proportional to the square of the distance at which the intensity is measured.  Thus, we write

From this we know that

I = k × 1 / d²

or

I = k / d²

where k is a constant.

Let us consider two distances d₁ and d₂ along with their respective intensities I₁ and I₂.  They will obey the equations

I₁ = k / d₁²

and

I₂ = k / d₂²

The ratio between these distances and, consequently, the ratio between the intensities can be derived by dividing the first of the above equations by the second.  This results in

I₁ / I₂ = d₂² / d₁²

which simplifies to

Solving for d₂, we have

Note that since guide numbers are distances and flash power levels are intensities, the above formula is perfectly descriptive of the relationship between flash power level and guide number.  We need only replace the variables with some which better represent the physical quantities of guide number and flash power level.  In this context, we can begin to rewrite the formula as follows:

What we have done so far is to replace d with GN (for 'guide number').  GN₁ is the published guide number of the flash unit, and GN₂ is the effective guide number we will calculate based on the flash power level setting.  The second step in our adaptation of the formula is to observe that I₁ / I₂ is a ratio between two intensities.  Furthermore, we can view the flash power expressed as a fraction as being an intensity ratio of its own.  It is the ratio between the power level at which the guide number was quoted by the manufacturer (full power or 1) and the power level in use.

Since GN₂ will decrease as the flash power level decreases, we can directly substitute the fractional power level like so:

That gives us the effective guide number in terms of flash power.  Next we need a formula for the effective guide number in terms of ISO.  We return to the following formula:

One way to express what this equation tells us is to say that the ratio of the published guide number to the effective guide number is equal to the square root of the ratio between the intensity at which the published guide number was measured to the intensity at which the flash will be used in a particular situation.  This relationship is a direct result of the inverse-square law.  Consider that for every time the flash power level is doubled or cut by half, the resulting photographic exposure changes by one stop.  This is also true of the ISO sensitivity.  For this reason, it is possible to treat the ratio of the ISO at which the published guide number was measured to the ISO used exactly as we did the flash power ratio.  We can substitute that ratio for the term I₁ / I₂.  Observing that as ISO increases, the effective guide number should increase, we perform the substitution in the following way:

ISO is the ISO sensitivity in use, and 100 is the ISO at which the published guide number is normally measured.  Now we have a formula for the effective guide number in terms of ISO.

Lastly we need a formula for use when both a flash power level other than full power and an ISO of other than 100 are used at the same time.  One way to go about this would be to perform the calculation in two steps, first calculating the effective guide number adjusted for flash power and then using that effective guide number in the formula for effective guide number adjusted for ISO to arrive at a second effective guide number adjusted for both flash power and ISO.  Let's proceed by restating the effective guide number formula in terms of flash power:

Assuming that we continue with our two-part strategy, we would then use the formula for effective guide number in terms of ISO in the following form:

Notice that the subscripts of the GN terms have been changed from 2 and 1 to 3 and 2, respectively.  This was done to exemplify the fact that in using the formula we would in fact be starting with the effective guide number given by the first formula (GN₂) to arrive at a second effective guide number (GN₃).  By performing a substitution of GN₂, we can write a single formula for the effective guide number adjusted for both power level and ISO as follows:

Below is a 3D plot of this equation:

Glossary