The guide number of a flash is determined as the F-number multiplied by the distance that gives a proper exposure. The guide number is different for different ISO sensitivities, but they are typically given for ISO 100, and when calculating a guide number for e.g. ISO 3200, you calculate:

$$\text{GN}_{3200} = \sqrt{3200\over 100}\times \text{GN}_{100}$$

and similarly for other ISO sensitivities.

Smartphone flash guide numbers are peculiar in that they are not only affected by the ISO number but also affected by the exposure time. For example if you have \$\text{GN}_{100,10}\$ for ISO 100 and 1/10 second exposure time, and want to know \$\text{GN}_{400,5}\$ for ISO 400 and 1/5 second exposure time, you calculate:

$$\text{GN}_{400,5} = \sqrt{10\over 5}\times \sqrt{400\over 100}\times \text{GN}_{100,10}$$

Let us determine \$\text{GN}_{100,10}\$ first and then other exposure times and ISO sensitivities can be calculated from that.

Let us fix the subject to be 1 meter away from the smartphone camera and let us consider an f/1 lens (fairly unusual for smartphones but F-number will be accounted for later), which would give a guide number of 1 meter. A field of view calculator gives image dimension of 1.2857 × 0.85714 units for a subject 1 unit away if the focal length is 28 mm and the image format is 3:2 (usually smartphones might have a differently shaped image sensor, but this shouldn't affect the calculations much because the crop factor is based on comparing the diagonal).

We also need to know about exposure value. For f/1 lens and 1/10 second exposure time, the exposure value is

$$\begin{align} \text{EV} &= \frac{\log\left({1.0^2\over 0.1}\right)}{\log(2)} \\ &= \log_2\left({1.0^2\over 0.1}\right) \\ &= 3.3219 \end{align}$$

Also we know that exposure value of 0 corresponds to 2.5 lux (from that Wikipedia article), so exposure value of 3.3219 corresponds to 25 lux.

If the smartphone manufacturer has modified the LED optics to produce an even light field for the field of view of the camera lens, the smartphone flash produces an illuminance of

$$\begin{align} \frac{50\,\text{lm}}{1.2857\,\text{m} \times 0.85714\,\text{m}} &= 45.371\,\mathrm{lm\over m^2} \\ &= 45.371\,\text{lux} \end{align}$$

So we have

$${45.371\,\text{lux}\over 25\,\text{lux}} = 1.8148$$

times as much light as we would need. This means guide number is not 1 meter as we initially assumed but rather

$$\sqrt{1.8148} \times 1\,\text{m} = 1.3471\,\text{m}$$

If the smartphone has a different ISO sensitivity (let's say 400), a different 35 mm equivalent focal length (let's say 35 mm), a different exposure tame (let's say 1/5 s) and a different brightness LED (let's say 70 lm) then we can take into account these corrections to the guide number:

$$\begin{align} \text{GN}_{400,\,5,\,35\,\text{mm},\,70\,\text{lm}} &= \sqrt{400\over 100} \times \sqrt{10\over 5} \times {35\,\text{mm}\over 28\,\text{mm}} \times \sqrt{70\,\text{lm}\over 50\,\text{lm}} \\ &\phantom{=\;} \times \text{GN}_{100,\,10,\,28\,\text{mm},\,50\,\text{lm}} \\ &= 4.1833 \times \text{GN}_{100,\,10,\,28\,\text{mm},\,50\,\text{lm}} \\ &= 4.1833 \times 1.3471\,\text{m} \\ &= 5.6353\,\text{m} \end{align}$$

Some comparisons for the guide number:

• A professional Canon EL-1 flash has an ISO 100 guide number at 24 mm of 19 m (this is measured, advertised is 27 m) and at 35 mm of 24 m (this is measured, advertised is 32 m)
  • To convert 24 mm guide number from 24 mm to 28 mm (assuming a perfect light distribution), you would calculate \$\left({28\over 24}\right)^2\times 19\,\text{m} = 25.9\,\text{m}\$
  • To convert 35 mm guide number from 35 mm to 28 mm (assuming a perfect light distribution), you would calculate \$\left({28\over 35}\right)^2\times 24\,\text{m} = 15.4\,\text{m}\$
• A very small Canon flash EL-100 has an ISO 100 guide number at 24 mm of 21 m (advertised; measured would probably be \${19\over 27}\times 21\,\text{m} = 14.8\,\text{m}\$)
  • To convert 24mm guide number from 24 mm to 28 mm (assuming a perfect light distribution), you would calculate \$\left({28\over 24}\right)^2\times 14.8\,\text{m} = 20.1\,\text{m}\$
• A typical DSLR built-in flash has an ISO 100 guide number of 12 m; not sure if this is measured or advertised, but if it's advertised the measured probably would be \${19\over 27}\times 12\,\text{m} = 8.4\,\text{m}\$ as advertised guide numbers are typically much larger than measured guide numbers.

Compared to these, the smartphone guide number of 1.3471 m is very poor and also to achieve that 1.3471 m, you need 1/10 second exposure time which means a shaken picture unless the smartphone is equipped with an image stabilizer, and also may mean you get motion blur unless the subject is perfectly still. So the smartphone flash is only useful at very near ranges in a very dark environment such as levels of light during streetlight periods, or in a very dark room. Also in macro photography in general home light levels the LED of a smartphone may be helpful.