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Aperture in crop sensors

There seems to be a lot of confusion when it comes to crop sensors and their equivalence to full-frame sensors.

Let's take the Sigma 30mm f/1.4 for Sony APS-C cameras as an example. The crop factor in these cameras is 1.5, which we can use to get the equivalent focal length of that lens: 30mm x 1.5 = 45mm.

The problem comes when we do the same with the aperture. Often, reviewers will apply the crop factor and say that a f/1.4 lens will behave as a f/2.1 lens on a full-frame sensor would. People get really offended because they think manufacturers are misleading the consumers with those numbers.

While that equivalence is true for the depth of field (smaller sensors have more depth of field, thus less bokeh), it is not true when it comes to light gathering.

A f/1.4 lens will gather the same amount of light (per inch) no matter the size of the sensor behind it.

In other words, you could share your exposure values of 1/50sec, f/1.4, ISO 100 with a smartphone shooter and someone using a large format camera. You all will get the same exposure.