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In chapter 27 of Feynman Lectures on Physics Vol. 1, section 7 on resolving power, Feynman states the rule for optical resolution:

two different point sources can be resolved only if one source is focused at such a point that the times for the maximal rays from the other source to reach that point, as compared with its own true image point, differ by more than one period.

setup

So the rule as stated by Feynman is

$$ t_2 - t_1 > 1/f $$

where $f$ is the frequency of light.

Feynman then continues:

If the distance of separation of the two points is called $D$, and if the opening angle of the lens is called $θ$, then one can demonstrate that $ \, t_2 - t_1 > 1/f \, $ is exactly equivalent to the statement that $D$ must exceed $λ/(n \ \text{sin} \theta)$, where $n$ is the index of refraction at $P$ and $λ$ is the wavelength.

However, what I get is $$ D> \frac{λ}{2 n \ \text{sin} \theta} $$ i.e. an extra factor of $2$ in the denominator. This holds when $D$ is sufficiently small (as is clear from the picture). If we define $t$ as the time to propagate from $P$ to $S$ or $R$ then by small angle trigonometry (not small in $\theta$!) we get that (omitting $c$ and $n$) $\ t_1 = t - D\ \text{sin} \theta \ $ and $\ t_2 = t + D\ \text{sin} \theta \ $, hence my result.

Am I making some trivial mistake? If not, how come this has not been spotted before?

N.B.: Feynman writes that $\theta$ is the opening angle of the lens, whereas in the picture it is depicted as the half-angle. Taking $\theta$ to mean the full angle $\angle SPR$ the resulting denominator should be $2 n \ \text{sin} \frac{\theta}{2}$.

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  • $\begingroup$ Is this the same as the Airy disk formula, see wikipedia. Is an aperture implied by angle theda? It could be an error in his notes, it's possible. $\endgroup$ Apr 11, 2019 at 13:38

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If you look at the pictures of this lecture and listen to the tape, it's apparent Feynman was winging it. He had a few pages of notes, weighted down under an ashtray on the lecture table, but he never looked at them. What he refers to as the "opening angle of the lens" is, in fact, half that angle, as can be seen in his blackboard figure. To the left of his figure you can see that he not only missed the factor 2 in the denominator, he put n*sin(theta) in the numerator! What he actually said was, "I haven't time to derive it here, but I'll leave it as a problem to see if you can figure it out, that this condition is exactly the same as this: that if the distance of separation of these two, on this point here, is called ugh (pause) d! (pause) and if the opening angle of the lens is called theta, then you can demonstrate that that is exactly equivalent to the statement that n times the sine of theta, where n is the index in this region - supposing that this is all built under oil or something with an index n - that n sine theta... (very long pause) [under his breath, to himself: ... must be, which way? ... then d lambda n sine theta...] D must exceed lambda n sine theta. (pause) D must be greater than that or you can't see it."

RPF's blackboard figure for FLP I:27, Fig. 27-9

This will be corrected. If 'label' will send me an email and tell me his or her name I will add it to the list of Contributors posted on our FLP Errata page.

Michael Gottlieb

Editor, The Feynman Lectures on Physics New Millennium Edition

[email protected]

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    $\begingroup$ The correction has been made in the online edition of FLP. $\endgroup$ Apr 12, 2019 at 10:15
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    $\begingroup$ You asked why no one has noticed this error before. I would wager it has been noticed, but when most people see an error in a physics book by someone like Feynman they assume the mistake is theirs. Those who convince themselves otherwise are a small minority, only a fraction of which will bother to write about it somewhere, and not all of those reports will make it to me. Besides, I've only been collecting and correcting errata for FLP online for the past 14 years, while the book has been published for 56 years. $\endgroup$ Apr 12, 2019 at 10:31

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