How would the size and brightness of the image formed by a pinhole camera change if the camera were made longer?

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Pinhole camera is a type of photographic camera where the role of the objective lens is take by a small hole. In contrast with a lens, the small home is afocal (it has focal distance at infinity) and parallel rays of light entering the camera will remain parallel after passing through the pinhole. This is also the principle on which this camera works, the light travels in a straight line through the hole. See the figure below for how the image is formed.

In the figure, the two triangles (formed by the light having as common point  the pinhole) and are similar.

`H/h = R/r` or equivalent `h = r*(H/R)`

and since `H` and `R` are constant we can say that the longer the camera (the further the image from the pinhole), the bigger the image is.

Now about the brightness of the image. The physical quantity the describes the quantity of light that passes through the pinhole is named luminosity (`L`) . Since the object does not change, the luminosity of the pinhole will be the same, regardless the length of the camera. The light flux density of the image is by definition the luminosity over the total area illuminated.

`Phi = L/S_("image") =L/(C*r^2)` 

Here `S_("image")=C*r^2` where `C` is a constant, from the similar triangles above (the image is similar on its height and also on its width with the object).

Thus we can say that the longer the camera the smaller the brightness (the light flux density) of the image (inverse proportional with the square of the camera length).

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