Exactly. And you have to decide if we're talking about focal length or angle of view. You can't freely interchange the two concepts, because they are two different things. You can compare one to the other, but within limits.
Sensor dimensions taken from: http://en.wikipedia.org/wiki/Full-frame_digital_SLR
FoV/Focal Length formulas: http://paulbourke.net/miscellaneous/lens/
(Using an 81mm lens here to be more accurate, 36/22.2 = 1.62 crop factor)
APSC sensor: 22.2mm width, focal length 50mm = 0.4369 radians = 25.0 degrees
Full Frame sensor: 36mm width, focal length 81mm = 0.4373 radians = 25.1 degrees
Focal length and Field of View are the same thing. The only reason they might not match in practice is because manufacturers don't quite tell the truth and the advertised focal length is slightly different from what they say (if it's 52mm, they're going to advertise it as a 50mm anyway)
A Nikon 1 10mm f/2.8 lens will have an equivalent focal length (read: angle of view) of a 27mm lens on FF, but it will be nevertheless a 10mm lens. Assuming transmission is the same, the exposure would be the same for this lens and a 10mm F/2.8 lens on FF.
A Nikon 1 10mm f/2.8 lens will have a pupil diameter of 3.57mm.
A 27mm f2.8 lens will have a pupil diameter of 9.64mm.
There are 2.7^2=7.3 times as many photons are coming through that 27mm lens as through the 10mm lens at the same f stop. Since the same proportion of those photons falling on the sensor is the same [same aspect ratios fitting in a circular aperture], the full frame sensor is receiving 7.3 times as many photons at the same f-stop.
The fact that one sensor is larger or smaller is completely irrelevant -- both sensors are covering the same proportion of the image circle (because we're talking about 35mm equivalent focal lengths here -- if you want to argue proportion of image circle covered then you need to do the comparisons using the same focal lengths -- see below)
The Nikon 1 lens can be smaller because the sensor is smaller. The total amount of light gathered by a FF will be more, but the amount of light hitting the sensor / surface of the sensor would be the same.
See above -- the amount of light (ie photons) hitting the sensor is most definitely not the same. The physical aperture size and the field of view are the only 2 things that control how many photons go through the lens.
You could mount the FF lens on the Nikon 1 and it would be the same - because the extra light will fall off the sensor.
Huh? A Nikon 1 sensor is 2.72x2.72 = 7.44 times smaller -- 87% of photons that would be hitting a FF sensor are being ignored. Assuming sensors are equivalent technology, at the same shutter speed the simple fact is that the Nikon 1 will have to amplify the signal coming off the sensor 7 times more than the full frame camera to get a proper exposure. How is this possibly equivalent?
That's why exposure is not affected.
More photons hitting the sensor = more electrons being excited = less amplification necessary to properly expose an image = less amplification of background sensor noise.
Note that I deliberately haven't mentioned ISO settings at all -- ISO settings are just calibrated labels to measure how much sensor signal amplification the camera needs to perform to properly expose a scene to a certain level, they are not an inherent property of the sensor itself. Go to dxomark and compare the SNR on a Canon 5D2 and a 60D -- the 60D is far noisier despite having a newer sensor, because ISO 1600 on a 60D actually means a lot more signal amplification than ISO 1600 on a 5D2. The ISO number itself is irrelevant, what is relevant is how much noise is amplified along with the useful signal.
If you now transpose this concept to this lens, you will understand that exposure-wise a f/1.8 lens is a f/1.8 lens disregarding of the system. This, again, is because f-stops depend on focal length and focal length is a parameter of the lens and not of the camera.
Yes, a f/1.8 lens is a f/1.8 lens regardless of the system. A f/1.8 lens on a full frame will perform better than a f/1.8 lens on crop with equivalent focal lengths though, and that's what the original question was.
Imagine that tomorrow I introduce a new camera system for amateur astronomers called "Super 100x35mm", sensor dimensions 360mm x 240mm, same ~4000x2800px resolution as a crop system. I stick 35mm-equivalent 50mm focal length lens on it, which for this system would be a 500m f/2.8. The pupil diameter of this lens is 178mm -- this pupil is 10 times wider than a full frame lens. According to the logic above, the fact that this pupil & sensor has 262 times more light gathering capability than a crop 32mm f/2.8 lens is irrelevant; "exposure-wise" all f/2.8 lenses are equivalent? Does this really sound sensible to you?
If that's the case, why are astronomy telescopes so wide?
- It's not to get a narrow FOV -- you could get http://pan-starrs.ifa.hawaii.edu/public/home.html FOV using a 650mm lens on a 35mm system.
- It's not resolution, you could just move the focal plane farther back to fit in more sensors -- eg 36cm could give you 1.7gigapixels with off the shelf components http://ambivalentengineer.blogspot.com.au/2012/08/argus-is.html
- It's because a wider aperture gives you more light = better low light performance
The original question was whether a crop camera with a given lens at f/2.8 performs the same as an equivalent focal length f/2.8 full frame. The answer is no, for equivalent focal lengths at the same f-stop the full frame sensor receives a lot more light and so gives less noise. In light of that, rs is right -- the lens might be nice for someone like me who only owns a crop camera but if you care terribly about low light performance then you should be on full frame.
Native sensor ISO? Let's not open *that* can of worms