Most photographers have started with a crop camera when digital photography was introduced. Only after a few years, the full frame cameras became affordable and that is the moment when all the discussions about the effect of depth of field and crop sensors started, often with the mentioning of physics laws and mathematical calculations. Let’s try to find out with some real world examples.
First of all, I think the whole discussion about how depth of field on a crop camera differs from a full frame camera is not really that interesting. I believe it is important for a photographer to understand how the depth of field behaves with his or her camera/lens combination, regardless of what kind of sensor the camera contains and not how much more or less depth of field that camera/lens combination has compared to another sensor size.
First of all, a crop sensor is smaller than a full frame sensor (which is obvious). Therefore, it captures only a part of what a full sensor would capture when using the same focal length. Thus, the picture with a crop sensor looks like it is magnified compared to what a full frame would show. In other words, it looks like the picture is shot with the focal length of the lens multiplied by the crop factor. This is visible in the example below, which shows what you capture with a full frame and a crop when using exactly the same focal length (120mm in this case).
The example above shows the difference of what a full frame sensor captures of a scene and what a crop sensor captures. If printed on the same size, it may look as if the photo with crop sensor used a longer focal length.
The second thing to realize concerns the mentioning of the focal length on a lens. This is always the real physical focal length, which has nothing to do with the sensor size. This automatically means the mentioned focal length of lenses that are made especially for crop sensor are not corrected for the crop sensor. Thus, a 17mm lens for a crop sensor has the same focal length as a 17mm for a full frame. The crop sensor will show only a field of view that is like a 24mm when compared to a full frame sensor (17mm multiplied with the crop factor).
Depth of field is affected also by the way you view a picture, how much it is enlarged, what the viewing distance is, and the pixel density on the sensor. But let us ignore this for the sake of simplicity. I don't want to make it more difficult than necessary.
To test the depth of field, I placed my camera on a tripod and shot a scene with full frame and crop, with the same focal length of 120mm and both with the same aperture of f/2.8. The distance to the subject was kept exactly the same. There are a few things that we see when these pictures are held next to one another. Of course, the 120mm acts like a 192mm lens when used with the crop camera, but the depth of field is exactly the same with both sensor sizes. This becomes visible when we enlarge part of the full frame photo to match that of the crop photo.
This shows the depth of field is not influenced by the sensor size when aperture, focal length, and distance to the subject are kept the same.
But wait, there is one problem. Under these conditions, we don’t have the same photo. It looks like we have used a longer focal length with the crop camera, which is due to the smaller sensor size. If we want to have the same composition, we have to change the focal length or the distance to the subject. Let us try both.
To get the same composition, we can reduce the focal length. This means we need a 75mm focal length (120mm divided by 1.6) to get the same angle of view. When we compare the photo between 120mm with full frame and 75mm with crop, we see the same photo emerge at least at first sight. But if we check depth of field, we see a difference; the photo with the crop sensor has a larger depth of field. The reason is obvious; it is due to the shorter focal length, which affects the depth of field.
Instead of using a shorter focal length, we can also change the distance to the subject. We need to get 1.6 times farther away from our subject to get the same angle of view with our 120mm focal length. When we compare this photo with the original full frame photo, we again see the larger depth of field, which is due to the larger distance from the subject.
To answer the question if depth of field is influenced by the sensor size, we can safely say yes, but it does so indirectly, because we are changing other parameters to end up with the same picture, focal length or the distance to the subject, which are the real reason why the depth of field is different between a full frame sensor and a crop sensor.