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I have been asked to recomend a camera to another person and they mentioned that they would like to do macro photography.
Assuming the same lens.
Is it best to recomend a full frame or a crop sensor camera in this case. Both alternatives are well within budget so that is not an issue.
My thinking is that the higher pixel density of most crop sensore gives an advantage, am I right in that reasoning or am I missing something?
For a given number of pixels, a crop sensor will indeed have a larger pixel density, thus a potentially larger resolution. But as the pixels are smaller, you'll have a bit more noise.
The real difference comes when you look at the size of the image on the sensor: you really like to have the subject fill the field of view. If you use a "real" macro lens which gives you a magnification of 1:1 (or more), at 1:1 the image is the same size as the subject. For a 1.5× crop sensor, a subject of 16 mm will take up the full height of the sensor, for a FF sensor, you'd need a 24mm subject. So the apparent magnification on screen (or uncropped print) will be larger for the crop sensor.
While not strictly related to the question, keep in mind that there's more to take into account when setting up for macro: when you are working at or close to 1:1 magnification (or more), handheld is getting tricky. So a good tripod is needed (and one that can go down to ground level). Also focussing becomes difficult, easiest is often to use a rail. Etc... All these extras have to be taken into account for the budget.
If assuming use of the same macro lens (same focal length) on both the FF sensor and the smaller cropped sensor, then the same lens will project exactly the same image onto both sensors. Same magnification, no difference at all in the lens image. If the lens is set to 1:1 magnification, the image on both sensors will be the same 1:1 magnification. Except of course the larger FF sensor has a larger frame around the subject, a wider field of view than the smaller cropped sensor. That is what cropped means, a smaller sensor with a more narrow view.
So if one sensor is 36 x 24 mm (FF), and the other is 24 x 16 mm (cropped smaller), then if the lens is set to 1:1, one scene view is 24 mm tall and the other scene view is 16 mm tall. However, that is NOT different magnification, it is simply different frame size, different field of view size. But any subject inside the frame will be exactly the same size in both (if both are the same lens at the same 1:1 setting, then both are same 1:1 life size).
A U.S. penny coin is 3/4 inch diameter, or about 19 mm diameter. So the full penny is easily shown in the 36x24 mm frame (larger field of view), but a bit of the 19 mm is cut off in the 24x16 mm frame. But the penny is the same 1:1 real life 19 mm size in both frames, because the magnification is of course the same from the same lens. It just cannot all fit into the smaller frame.
You may have to stand back a bit further with the smaller cropped sensor to see the entire subject view in the smaller frame again, which then would be less magnification (perhaps 1:1.5 instead of 1:1, to see the same view). This is not a problem, you simply stand back 1.5x further to see the same scene as the FF sensor sees. In fact, the ability to stand back a few more inches is usually a plus for macro work. But you do always have to enlarge the smaller cropped images more to view them at the same size again.
Any later cropping of the larger FF frame to match the smaller cropped frame view will lose a bit more than half of the overall pixels (which is sometimes counterproductive). Then both smaller images (from cropped full frame view, or from smaller cropped sensor) have to be enlarged more to view at same size again as the original FF sensor image.
But in real world, we simply use either sensor in the way it should be used, to frame the subject in the desirable way.
