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The Relationship Between Aperture and Focal Length
Article: The Relationship Between Aperture and Focal Length
Very interesting and well thoughtout article, Punit! Great read. 


Yeah, I've actually linked to this article in a couple other places on the net.



This is currently the resource the gets the most "hits" on our site thus far!



This article is completely wrong.
Unfortunately, this article is completely wrong. The intensity of light is only inversely proportional to the square of distance when the light is emitted from a point source. This does not apply to the light inside a camera lens.
The correct explanation is this, assuming you are photographing an object far away: Each point on the object emits (or reflects or scatters) light. The amount of this light which is admitted by the lens is proportional to the square of the aperture diameter. When you use a longer focal length, though, you spread the light from the same area of the object over a larger area of film. If you double the focal length, the image is twice as big, so the light from that part of the object is spread over four times as much film. If you increase both the focal length and the aperture, keeping the f/# constant, these effects exactly cancel, and the image brightness is unchanged. This is discussed in Optics, 2nd ed., by Eugene Hecht, AddisonWesley (1987). Ch. 5.3.3, page 152. Note that this doesn't quite work if the object is not very far away. The f/# correction is slightly off for objects that are close to the camera. I'm not sure if this is of practical significance in photography, though. (I'm an optical physicist, not a photographer.) Another quibble: contrary to the article, f/# is not the ratio of focal length to aperture diameter, if by aperture you mean the physical opening in the iris inside the camera. This would be true if the iris were in front of the lens. Since there are optical elements in front of the iris, one has to correct for the magnification of these elements. The correct statement is that the f/# is equal to the focal length divided by the diameter of the entrance pupil of the lens. The entrance pupil is the image of the aperture formed by the optics in front of it. What is true, though, is that the fnumber is proportional to focal length divided by aperture diameter, so if you double the aperture diameter, the f/# decreases by half. 


To say that this article is wrong and to give a correct explanation is similar to saying that the equation 2 + 2 = 4 is wrong and the correct equation is one added four times will give four.
Quote:
and at the same time the intensity of the light has also reduced. Its the same amount of light, so if you spread the same amount of light over a larger area the intensity is reduced, that is where the inverse square law comes in. ..........................When you use a longer focal length, though, you spread the light from the same area of the object over a larger area of film ................. When you use a longer focal length, the light has to travel more and spread more, without travelling more distance it cannot spread more. and so the inverse square law rightfully states that "The intensity of the light is inversely proportional to the Sq of the distance it travels.” ...............If you increase both the focal length and the aperture, keeping the f/# constant,.................. If you increase the focal length, keeping f/# constant, the size of the aperture will automatically increase. ...........these effects exactly cancel, and the image brightness is unchanged............ The image brightness is unchanged, and the mathematical equation in the article using the inverse square law just proves this fact. ..............This is discussed in Optics, 2nd ed., by Eugene Hecht, AddisonWesley (1987). Ch. 5.3.3, page 152.............. Though I have not yet read this book, ordered it from the library, will get it in a day or two, I am sure this would have been discussed there and the explanation would be right, but I dont see how this explanation proves that the article is wrong. Am I missing something somewhere. To me it appears that the explanation is the representation of the mathematics discussed in the article in generic layman's language. ...........Another quibble: contrary to the article, f/# is not the ratio of focal length to aperture diameter, .............. Do I even have to bother to respond to this? 


Aperture and focal length.
One of the differences between physics and mathematics is that it is possible for an equation or formula to give the right result, and yet not be a correct physical description of the phenomenon. The main problem with your explanation is that, both in the article and your reply, you assume the light between the lens and the film is spreading out as it propagates. This is not the case. The light is focused onto the film. In particular, the light rays that leave a given spot on the object being photographed converge to a single spot on the film. They do not spread out as they propagate between the lens and the film. If they did, no image would be formed. Nor does the light necessarily get less intense as it travels between the lens and the film. The inverse squared law is simply not the correct model for light propagation inside a camera. It is only by chance that applying it gave the correct answer.
I'm not sure why you felt my comment about the definition of f/# was not worth bothering with. Do you disagree, or do you just not see the distinction as important? The assertion in the article that "f/8 means an aperture whose diameter is f divided [by] 8" is incorrect, at least if "aperture" refers to the size of the physical iris opening as opposed to the size of the lens' entrance pupil. While this is a sideissue with respect to this article, I brought it up because it appears to be a common misconception among photographers, that the focal length divided by the fnumber gives the size of the physical opening of the iris diaphragm. This is not the case. 


I have a feeling this is going to be a never ending discussion.
From your first post: ...........When you use a longer focal length, though, you spread the light from the same area of the object over a larger area of film.......... How does it spread without travelling a larger distance and from your second post: ..........They [light rays] do not spread out as they propagate between the lens and the film............ You are contradicting yourself, and I have not pulled two statements from your previous two posts out of context, to show you that you are contradicting yourself. Read on... Now from your first post: My comments are in red Each point on the object emits (or reflects or scatters) light. The amount of this light which is admitted by the lens is proportional (inversely or directly, I am sure you mean directly) to the square of the aperture diameter. When you use a longer focal length, though, you spread the light from the same area of the object over a larger area of film.(How does the light spread if it travels the same distance, longer focal means the light has to travel more and so it spreads and that is in accordance with inverse square law. And that is precisely what the article shows, mathematically) If you double the focal length, the image is twice as big, so the light from that part of the object is spread over four times as much film. If you increase both the focal length and the aperture, keeping the f/# constant, these effects exactly cancel,(By exactly cancel you mean (assuming the focal length was doubled) that the amount of light that came is in 4 times more(remember directly proportional to square of aperture diameter) and the amount of light on the film is 4 times less, as it spreads over an area which is 4 times more, and again this is what is shown mathematically in the article) and the image brightness is unchanged. and now for "THE ANOTHER QUIBBLE" regarding the f/# Come on now, if you are discussing with me, I assume that you know your stuff pretty well when you say, "...If you increase both the focal length and the aperture, keeping the f/# constant, these effects exactly cancel,..." and you say "..........The amount of this light which is admitted by the lens is proportional to the square of the aperture diameter..........." These two statements of yours tell you that f is directly proportional to Aperture, you double f, you double the Aperture diameter(and increase area by 4 times, area of is cirlce PI*R*R ). and anyone will tell you that the size of the Aperture diameter is focal length divided by fstop. 


Great discussion here!!
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