Light and the Inverse Square Law

At some point in your photographic life you will inevitably come across something called the Inverse Square Law (ISL), which relates to light fall off with distance and how that affects exposure. A lot of photographers are put off by this mathematical construct, and while the physics behind it is relatively complex, its quite easy to understand and apply.

What is the Inverse Square Law?

The inverse square law is a general law in physics that applies to electromagnetic radiation, which includes light, and also applies to acoustics, electrostatics and gravitation.

In essence, when applied to light, the law states that the intensity of light from a light source falls off, or is reduced, with the square of distance.

Mathematically, intensity is proportional to the inverse square of distance, which can be expressed as:

Intensity ∝ 1/distance²

(∝is the symbol for “proportional to”).

So what this means is that if you measure the light intensity at 1m from a light source, and get a value of, say, 100, then if you move to 2m from the light source, you will get a reading of 25, i.e. ¼ of the value at 1m (100) – since 1/2² = 1/4.

Thus light fall off is stronger than you might have expected – some people might have thought that if you double the distance, you half the light, i.e. 50 in this example, but in fact you only get a quarter of the light intensity.

Its actually quite easy to understand why if you visualise a section of light emanating from a source as a cone. At the light source, the cone cross-section containing all of the light is small, leading to a high intensity. As the light travels further from the source, the same amount of light is spread over a larger cross-sectional area, leading to lower intensity. The increase in area of the cone cross-section with distance is related to the surface area of the complete sphere, which is proportional to the square of the distance (sphere surface area = 4πr²).

Relevance of the Inverse Square Law to Photography

The relevance of ISL to photography is that it shows why the distance between your subject and your light source is important in terms of light intensity, i.e. exposure (as well as in terms of quality of light – see Hard Light vs Soft Light).

For example, in the studio, ISL gives us a way of understanding how a main light will affect the relative illumination of the subject and the background, i.e.the light ratios, and so how to position them relative to the light source to achieve a particular relative light ratio.

This is how you can make a white wall behind a subject look white, grey, or even black while still properly exposing the subject.

The ISL is also important to portrait photographers – if you place a light close to your subject, the relatively high fall-off across their face can give dramatic results, because the relative distance between one side of the face and the light, and the other side of the face and the light is proportionally higher.

The other facet of lighting that ISL helps us with is when shooting multiple subjects. If you have two subjects standing 1m apart, and 1m and 2m from a light source respectively, according to ISL the person 2m away will be illuminated at 1/4 the intensity of the person at 1m away, i.e. the illumination ratio between the subjects is 4:1.

That is quite a difference and means that it will not be possible to expose for both subjects if you want them to be evenly lit.

If however you move the light source away, keeping the subjects where they are at 1m apart, so that they are now 10m and 11m from the light source respectively, the relative difference in distance (1m) between them and the source is much less. This means that their illumination relative to each other is closer making it easier to expose both adequately – in this example the illumination ratio is approx. 1.2:1 (of course for this to work the exposure will need to be adjusted, either by changing the aperture, slowing the shutter or increasing the power of the light, or a mixture of all three: at 10m away vs 1m away the subject would be receiving 1/100 of the light, which is somewhere between 5 and 6 stops).

OK, the maths there might have been a bit daunting, and becomes even more so with even more subjects, but the thing to takeaway from it is to try to even out the relative distances of each subject from the light source if you want their illumination to be similar.

Limitations of the Inverse Square Law in Photography

The obvious thing that might limit the application of the ISL in photography is that strictly the law applies to single point light sources. Very few of the light sources we use in photography are single point. Although for most practical purposes the law is close enough and ISL can be used.

Modifiers

Fundamentally the whole purpose of light modifiers is to change the nature of the light source. For example soft boxes generally soften light by making the light source appear bigger (see Hard Light vs Soft Light); parabolic reflectors and beauty dishes create a focal area where the light is more intense, and fall-off is accentuated outside of this area; Fresnel lights produce near parallel light.

While each point on the light modifier could be considered as conforming to ISL, when taken together the result is more complex: for example there could be a near field where fall-off is minor, a far field where ISL does apply, and in the middle field some kind of combination.

Multiple Light Sources

When there are multiple light sources used, this will obviously have an effect on fall-off depending on relative placement. However, ISL can be applied to each light source individually, and then a combination of the relative fall-offs made to give an overall guide.

Natural light

Light from the Sun hitting us on Earth is considered to be parallel light. This is because the Sun and the Earth are so far apart.

The Inverse Square Law does not apply to parallel light.

A lot of photographers write about using the inverse square law in natural light portrait photography to achieve a higher exposure on their subject than the background, but when you analyse what they are doing they are actually making use of shadows, that is some object is causing a shadow on their background and they position their subject outside of this shadow area in the light.

The Sun is 149.6 billion meters from the Earth. So even if you were to apply the ISL and assume your background is 20m further away (from the Sun!?), the difference in relative distance from the Sun to your subject and background is miniscule:

20 / 149,600,000,000 = 1.34xE-10 (0.000000000134)

The above applies to direct illumination by natural light – if the light is attenuated, for example through a window into a room, the ISL can be applied to estimate fall-off by treating the window as the light source.

Conclusion

The Inverse Square Law is useful to know, because it explains why light fall off may be more drastic than we might expect, and thus give us hints on where to place elements of our scene. Its also good to know how ISL can be affected by light modifiers,

The best way to know for sure is to use a light meter to check the incident exposure.

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