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In order for them to frame the same scene, the FX must be closer to the subject/scene. This has a dramatic impact on the intensity of the light wave incident at the camera location.Set up an FX camera with a 50mm lens at f/2, and an APS-C camera with a 35mm lens at f/2, and set them to frame the exact same scene with the same shutter speed aperture and gain.
The FX camera will admit 2.25x the number of photons as the APS-C camera.
More photons means more signal. It means more signal in relation to the electronic noise, which is relatively fixed. This means a higher signal-to-noise ratio.
This is the reason that noise measurements are made /per unit area of the sensor/. You will find that total noise is mostly fixed by the size of the sensor, regardless of how many pixels the sensor carries.
Also be advised that photon theory (partical theory) and Inverse Square Law are not compatible because the Inverse Law applies to Wave Theory.
My firm belief is that when a scene is framed equally at different distances, that the light is identical except that the intensity is degraded at distance and that the degrading is very significant.
Of course when I say equally, I am aware that they are approximately equal. For my shooting style it is 6 inches above the model's head and 6 inches below her feet. So I am fully aware of the slightly different incident light caused by wider angles at shorter distances, but otherwise equal.
So FX receives more intense light. Current competitive designs mean that the FX also delivers less MP. Therefor the "Signal" is not equal. There is more noise when 24MP are delivered than 18MP. So what do you mean when you say s/n ratio?