Aperture vs sensor size

[KeithB]

Not a very clear topic indicator, but...

I know that f-stop is the ratio of the Aperture to the focal length.

So if you have a low aperture and long focal length you have a very wide piece of glass. But, F/Stop does not change when I use an EF lens on an EF-S camera, though I "waste" some of the image circle.

So, I guess the basic size of the lens is based on the aperture for a given image circle. I *could* put my S-90's lens on my APS-C camera, but while the exposure would be correct, it would only illuminate a small part of the sensor, correct?

Is there a formula for Aperture size to image circle?

[klindup]

A very good question Keith.  I had not thought about this before.  On the face of it given crop factors of 1.5 or 1.6 implies that on an APSC sensor the effective aperture is one stop less than the indicated aperture. So when you think that your exposure is 5.6 it is really closer to f8.  Clearly this is not the case so is it a case that a smaller sensor requires a smaller number of photons to create an image?  In the case of APSC versus full frame sensors 1.5 times less?  I am sure that Bob has an explanation.
Ken

[Bob Atkins]

It's the image brightness on the film/sensor plane that counts. It doesn't matter whether the sensor sees all of it or just a part if it, the "number of photons per sq.cm" is the same, so the exposure is the same.

An analogy would be rainfall. If it's raining at 1" per hour, it doesn't matter whether you collect water in a 12" bucket or a 3" cup, you still get 1" per hour.

The larger sensor does capture more light, but it's spread out over a larger area, so the exposure (which depends on light per unit area) is the same as for a smaller sensor which captures less total light, but over a smaller area.

Nothing changes when you use a AFF lens on APS-C. The actual focal length is the same and the aperture is the same. The only thing that actually changes is the angle of view, which can be (and usually is) though of as a change in "effective focal length relative to full frame". Depth of field changes only because of the way it's defined, which is based on a certain print size, not a certain degree of magnification of the original image.

[KeithB]

I guess what I am asking is:
Is there a formula to determine the "correct" size (i.e., size of the image circle) of a set of optics? For example, for my S90 I have a really tiny lens, while my Rebel needs a larger lens, even for the same F stop. To make it concrete, the S-90 has a 6-22.5 mm zoom with an fstop of f/2-f/4.9. An equivalent lens for my rebel, would be huge - probably the size of the EF-S 10-22. Now the EF-S 10-22 should work fine on the S-90, but the S-90 lens would not work on the rebel. Is there a simple formula to determine how "big" a lens needs to be to work on a given sensor size?

[Bob Atkins]

Not really. Obviously the larger the sensor the larger the image circle that is required to cover it. However the size of the image circle depends on the lens design. A 20mm f2.8 for a full frame sensor would likely be physically larger than a 20mm f2.8 for an APS-C sensor, but you can't really relate "front element diameter" to "sensor size" or "image circle" in any direct, linear manner. They are not directly proportional. It depends on the lens design and the diameter and position of internal aperture stops (and/or lens elements). You can use smaller diameter lens elements for smaller sensors which require smaller image circles. In theory this should lead to cheaper lenses...

If you scale focal length with sensor size, then a 50/2 APS-C lens needs a front element at least 25mm in diameter, whereas the "equivalent" FF  80/2 lens needs a front element at least 40mm in diameter. For medium and telephoto lenses this holds true, but for very wide angle lenses you can't calculate front element size from focal length and aperture. A 10mm f4 lens is going to be more like 70mm in diameter than the 2.5mm you'd calculate just from dividing the focal length by the aperture.

[KeithB]

So the answer is: its complicated.

I guess you can make the image circle any size you want by playing with the lens->sensor distance, which might be the piece I am missing.

[Bob Atkins]

It's more the intrinsic lens design. You can make a 100mm lens cover APS-C, FF, Medium format or 4x5 and in each case the lens will be the same distance from the film/sensor (though it's a telephoto on APS-C and a wideangle on 4x5). The 4.5 version will have considerably larger diameter optics than the APS-C version though, and will cost a lot more. It also won't be as sharp in terms of lp/mm resolution as the APS-C version, though of course the larger format makes up for that when you make a print of a given size.

It is, however, easier to make a longer focal length lens with a large image circle than it is to make a shorter focal length lens with the same image circle.