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Refraction, Fermat's Principle and Lawnmowers

  1. Apr 17, 2007 #1
    According to Fermat's principle the path taken between two points by a ray of light is the path that can be traversed in the least time. So if the ray is refracted and if you take two points, one from before and one from after the refraction, the path taken between them was the one that takes the shortest time.

    This fact can be explained by Quantum Electrodynamics, which states that any particle propagates over all possible paths and in the end all paths except extremal cancel each other out.

    What I am wondering though is the following: you can use various analogies for Fermat's principle, e.g. a lawnmower, moving from a sidewalk to grass: http://web.archive.org/web/20051226...5/LectureNotes/LightOptics_files/image028.jpg

    But does Quantum Electrodynamics also apply to a lawnmower? I am not an expert regarding QED, but a lawnmower isn't exactly a quantum object like a photon, still if you move it from a sidewalk to grass and take two points, one from before and one from after the bend, the path taken between them was the one that takes the shortest time. Why doesn't the bend happen in any other way?

    Is there a different explanation for it? But if there is one, isn't it strange then that different explanations lead to the same behaviour in different cases (out of the many possible paths, the one that takes the shortest time is taken - in the case of the refracted ray of light and in the case of the lawnmower)?

    Maybe someone can help regarding this issue as I am wondering about this for quite some time now.
  2. jcsd
  3. Apr 17, 2007 #2
    Fermat's principle do not says that the path is a minimum. It says that the path is an extremum. There are a lot of cases where the extremum is a maximum. Take the case of a ray that goes between the two points over a German Ü.

    I do not understand the analogy with a lawnmower. A lawnmower do not obeys QED, not even reflection angles and sometimes even their owners.
  4. Apr 17, 2007 #3
    Yes, it's only an extremum... okay. Anyway, in the case I was talking about it's a minimum.

    To explain the case of the lawnmower, well, just take a look at the picture:

    You can regard the axle between the two wheels as a straight line and the two different undergrounds as two different media on which the lawnmower has a different speed (it's slower on the grass than on the sidewalk).

    Now, when the lawnmower moves forward, at first only one wheel touches the grass and gets slowed down, while the other wheel is still on the sidewalk - so as a result the axle turns to one side and the path of the lawnmower is bent. Now you can take two points, one from before and one from after the bend and the path taken between them will be the one that takes the shortest time.

    This is the same result as with a ray of light going through two different media with different refrective indices, which is then bent in the same way (which in that case can be explained by QED).
  5. Apr 17, 2007 #4
    Now I understand what you mean with the lawnmower. It is one illustration of the Huygens principle in a rather twisted way. I prefer the old one with front waves and wavelets. And I think that if you had read the old explanation, you would have had fewer problems.

    Last edited: Apr 17, 2007
  6. Apr 18, 2007 #5
    So as I see it Huygens principle can also explain the refraction of light - but after reading the Wikipedia article I'm not quite sure how Huygens principle can be combined with QED. Does light advance through space like the wave fronts in Huygens principle or does it advance over any and all allowed (unobstructed) paths as in QED?

    I'm also still unsure which principles apply to the lawnmower. The Wikipedia article says the following concerning QED:
    All other objects - does this include non-quantum objects like the lawnmower?
  7. Apr 19, 2007 #6
    Light advances as modelized by Huygens principle. This principle not only explains well refraction and reflection, but also diffraction and interferences.

    In QED it has been necessary to add that a photon behaves as if it takes all possible paths. This was the only way to explain how a single photon can give interference patterns (well, not a single photon, but a lot of isolated photons).

    The dual behavior particle-wave is common to all particles and objects. But if diffraction by electrons is "easy" because electrons with a kinetic energy of 100 eV have an associated wavelength of the order of the atom separation in a solid. For neutrons that are 2000 times as massive, you need to freeze them at liquid helium temperature to obtain diffraction.

    Your lawnmower has an associated wavelength 10^28 or 10^30 times smaller that that of an electron. You will have a hard time to find a smooth enough surface to try to make it rebound.

    I do not think that Huygens principle should, can or must be combined with QED. In physics we have a lot of models and we take each time the good model to explain an aspect of nature. When you work with light, you take the wave model for some things and the photon model for others. Or even the two at the same time to explain some experiences. But you do not use QED for light. Each time you choose the adapted model.

    Do not forget that, after all, they are just models and not the absolute truth or the reality of nature. And each model just describes an aspect of things. This is true for all models: from Newton laws, to QED, Huygens principle, or strings theory. These are just models, not the Revealed Truth. Only Gods, priests, psys and economists know the Verity (or think they do).
    Last edited: Apr 19, 2007
  8. Apr 19, 2007 #7
    So it seems that QED isn't absolutely necessary to explain the Huygens principle. It is only needed to explain the behaviour of single photons.

    Yes, of course they are only models, but it's the best we currently have to try to explain and understand how the universe works - and that's what I'm trying to do. :wink:

    I have looked around a bit at Wikipedia and read about so-called variational principles of which Fermat's principle is one (the principle of least time or extremal time), another being the principle of least action. Apparently historically it was believed that God had created the world in an optimal way, which explains these principles.

    I am not completely sure about this, but I think that these principles could rather be results of other functionalities or relations, e.g. the least action principle could automatically follow from the principle of acceleration by forces. I am not quite sure why the extremal time principle must automatically follow from Huygens principle, but maybe there are some mathematical reasons for it or something else...

    An interesting article is this one: http://www.csicop.org/si/2006-04/quantum-mechanics.html
    If I understand the author correctly he applies Feynman's view of QED and the path integral to all objects and explains the least action principle by this, completely abandoning acceleration by forces. He uses some weird analogies however... I'm still a bit confused though - maybe I should read Feynman's "QED: The Strange Theory of Light and Matter" in order to get a more complete picture of those ideas...
  9. Apr 19, 2007 #8
    Yes, you should read Feynman's "QED: The Strange Theory of Light and Matter" It is a delicious little book, and I am sure that you will enjoy it.

    I'm no sure of the theoretical basis of least actions principles. Nor of Huygens principle or Lentz law. Or saying that "systems tend to evolve to the lower potential energy status". I think that all this are "experimental laws". That is, things that work but that can't be demonstrated.
  10. Apr 20, 2007 #9
    1) Why does the behaviour of the lawnmower resemble the
    behaviour of light?

    (i) If we assume that light is taking the path with the minimum
    time period, we can make the calculations here and derive Snell's refraction law.
    (ii) If we decide that the lawnmower takes the path with the minimum
    time period, we can make the same calculations from above and derive Snell's refraction law.

    Let us state the following
    Theorem: Whenever we take the path with minimum time period, we can observe refraction (Snell's law). This is shown mathematically on the website above.

    So you see, the resemblance occurs because WE set a condition (minimum time) upon the lawnmower. In other words, WE created the
    resemblance by letting the lawnmower imitate the light's behaviour. (The resemblance does not occur anymore if we don't set the condition.)

    2) Can QED be applied to the lawnmower

    (i) QED can be applied to light. Why? Because it predicts the light's behaviour (refraction).
    (ii) QED can NOT be applied to the lawnmower. Why?
    Because it does NOT predict the lawnmower's behaviour.
    The lawnmower could for example also walk in a straight line from A to B
    (without "refraction").

    3) Why doesn't the bend for the lawnmower happen in any other way

    (i) If we set a condition for the lawnmower (take the path with minimum time), then indeed the bend doesn't happen in any other way.

    (ii) But if we don't set the condition for the lawnmower,
    then the bend CAN HAPPEN in another way.
    The lawnmower could for example take
    - path a) walk in a straight line from A to B with no bend at all
    - path b) walk along the sidewalk horizontally to the right, until above point B
    and walk vertically down to B (90° bend)

    Both in a) and in b) the bend does not happen in way that fulfills Snell's law.


    I hope you recognized why, or better WHEN the behaviour of light and the and that of the lawnmower resemble each other. The resemblence only occurs in a special case, namely if we set a condition upon the lawnmower.
    Last edited: Apr 20, 2007
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