Wednesday, November 12, 2008

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

4 comments:

:: J o h n :: said...

WALAO.... super confusing ler this is what is behind all the math. I just take what was told to me lAH haha

Templar Tan said...

@john
Haha... nvr mind, just remember the last conclusion only lar!
haha

Master TM said...

Cold sweat breaks out from my forehead reading the maths..took me back to schooldays where bad dreams are all about maths..muahahaha..

Templar Tan said...

So, Master TM, did you able to get what I wrote ? LOL