50mm on APS-C = 75mm on FF ?

Let's talk the Physics behind the lens, shall we ?

Ok, I have an APS-C sensor camera, and I have a 50mm lens. My 50mm lens is an old full frame (FF) lens. Meaning, it's made and used before the digital sensor camera was introduced into the market.

I know that APS-C sensors are smaller than the FF sensor. How small? I'm an engineer, hence I read numbers to comprehend facts. The size of a FF sensor is 35.9mm by 24.0mm so to give an area of 861.6 sq. mm. And the size of an APS-C sensor is 23.5mm by 15.6mm giving an area of 366.6 sq.mm. Thus, APS-C sensor is in fact smaller than the FF sensor.

Hmm... sound stupid I am, right ? LOL. Well, I have time to kill, hence, just talk some crap ! LOL

Anyway, when we talked about mounting an FF 50mm lens onto the APS-C DSRL camera, the correct focal length is no more 50mm, but to multiply a crop factor of the APS-C sensor. Hence, in my case, my 50mm lens is actually a 75mm lens on my trusty A700 camera.

Sound complicated, eh ?

Well, again, I'm engineer and I speak in numbers. To understand more about this focal length change from 50mm to 75mm, I turn to my STPM Physics!

Oh crap, I had returned all my physics back to my teacher after I left school about 10 years ago. Refer back to the text book is a wise move to refresh my deep buried knowledge of optical physics.


This Advance Level Physics textbook written by Nelkon&Parker is damn boring if compared to the University Physics textbook. Well, what could I expect from a textbook that is written and published by a Brit ? LOL


Ok, turn to the page where the textbook talk about the physics of optic. There is a thin lens topic with the the Lens Equation is being derived. Not to bored myself, I skip the derivation part, and straight search for the thin lens equation, that is

The Lens Equation !

1/v + 1/u = 1/f

where v = distance of object from lens axis, u = distance of image from lens axis and f = focal length of the lens.

So, I have my equation. Then how to show that a 75mm FF lens is actually 50mm APS-C lens?

Not to complicate the whole thing, I assume my lens is a single lens only. Hence, my focal length, f is 50mm.

Say I have an object of 600mm in height. Say to cast the image onto a FF sensor, the size of the image will be 24mm. Hence, the magnification factor, m = 24/600 = 0.04.

By definition, the Magnification Factor, m is v/u or size of image/size of object.
Therefore,

m = 0.04 = v/u
v = 0.04u

So, in the lens equation, rearrange, I got this

1/0.04u + 1/u = 1/50

Solve it, I got the distance of object is u = 1,300 mm.

These numbers say : when an object of 600mm in height is placed at 1,300mm away from a 50mm lens, the image size is 24mm on a FF sensor.

So, what happen if the same object is cast onto an APS-C size sensor?
The size of the APS-C size sensor is 15.6mm. Hence, the magnification factor is 15.6/600 = 0.026.

Since the focal length of the lens doesn't change, hence the f = 50mm. Solving the equations similar to the above give this

1/0.026u + 1/u = 1/50
u = 1,973.07mm

So, this number tells me, if I put my object at 1,973.07 mm away from the lens, the image size will be at 15.6mm on a FF sensor. But, the actual object is actually placed at 1,300mm away from the lens. Hence, the actual lens focal length that give the image size of 15.6mm is

v = 0.04 x 1,973.07 = 78.92 mm
1/1,973.07 + 1/78.92 = 1/f
f = 75.89 mm

Hence, as conclude, when a 50mm FF lens mounted on an APS-C camera, I actually get a 75mm zoom lens instead.

So, do you understand what I wrote so far ?

I admit it sounds confusing. To sum up everything, when we mount a FF lens onto a APS-C camera, we get extra zoom by 1.5x factor.

Got it ?

No? Never mind, because I started to confuse also ! LOL