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How does the power of a lens affect its ability to focus light?

  1. Nov 16, 2014 #1
    So I've been doing some experiments, using bottles filled with various liquids and timing how long they take to burn paper which is placed at the focal point (where the light was focused). I've noticed that using denser liquids with high refractive indexes lower the focal length, and thus increase the lens power. This causes the bottle to be able to achieve a faster burn. Specifically, using glycerol in comparison to water in the same bottle allowed me to halve the time it took to achieve a burn.

    My question is, why? I've searched frantically over the net, and have found two answers I need explanation and elaboration on.

    1. The inverse sq Law - this means that light being focused over a longer distance will diminish in intensity. However, I do not believe that this applies to this context.

    2. A larger focal length means a larger focal spot, thus reducing intensity of the light - I simply don't understand this. Don't all convex lenses converge light onto one small spot? Why does the size of this spot change? Isn't this related to aberrations/refractive errors instead?

    Please, if possible include sources in your responses.
     
  2. jcsd
  3. Nov 16, 2014 #2
    http://en.wikipedia.org/wiki/Focal_length

    The focal length of an optical system is a measure of how strongly the system converges or diverges light... A system with a shorter focal length has greater optical power than one with a long focal length; that is, it bends the rays more strongly, bringing them to a focus in a shorter distance.

    You have all the physical facts you need, since it is optics it is obvious to "see" where the power lies. What do you see when you look through the lenses and focus an image? I suggest you look at text on paper and report your observations...
     
  4. Nov 16, 2014 #3
    Unfortunately I haven't been able to look through the lens and focus a very good image (bottles aren't very good), though I can tell it magnifies to an extent. I've been able to focus an image of the outside (window) however, and the image is inverted on both axes.

    Still, I don't understand how the focal length affects it's ability to converge more or less light. How does focusing light at a closer distance focus more light than at a farther distance? Shouldn't all light that enters the lens be focused at the focal point?
     
  5. Nov 16, 2014 #4
    Since your lenses are crude you can't "clearly" see what I was trying to point out, but when you focus the lenses, for instance on a character of text, if the optical power is greater, the text "appears" larger. When the focal length is longer, the text looks smaller (farther away). What does this imply about the inverse square law?
     
  6. Nov 16, 2014 #5
    If it were a concentrated beam of light (like a laser) in a vacuum, would it matter how far away the target was?
     
  7. Nov 16, 2014 #6

    mfb

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    It can matter, but for laser beams the distances where this becomes relevant are huge.

    The sun is not a laser beam - it has a finite size. If the focal length is too long this becomes relevant. You might be able to focus the light emitted by "a point" on the sun to "a point" on paper - but not all points from the sun to the same point on the screen.
     
  8. Nov 16, 2014 #7
    Fact checking #1 first...
     
  9. Nov 16, 2014 #8

    Drakkith

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    Number 1 doesn't apply. The inverse square law only applies to light that is diverging in a spherical wavefront. A good example is a laser vs a light bulb. Laser light doesn't diminish in intensity according to the inverse square law because it has been focused into a beam. But light from a light bulb does obey the law because it spreads out in a spherical pattern, not a beam like a laser.

    Number 2 is the correct reason. Light is an electromagnetic wave, and you cannot focus a wave down to a single point. Instead, the wave is focused down to a finite sized "spot". The underlying reasons why the spot size is different for different power lenses is actually kind of complicated and I'm not sure I can explain it very well, but it has to do with how different parts of a converging wavefront interfere with each other. With longer focal lengths, the different parts of the wavefront will constructively interfere over a large area at the focal point, which spreads out the energy and forms a larger spot. With a shorter focal length, the different parts of the wavefront constructively interfere in a smaller area, which leads to a small spot.

    When talking about images, the physics lead to a smaller image at the focal plane when the focal length is shorter, which focuses all of the energy into a smaller area, quickly heating up the paper compared to a longer focal length which has a larger image at the focal plane.

    See the following link for a very good (but also very in depth and complicated) source for understanding optics. It's geared for telescope optics, but the same principles apply to all optics.
    http://www.telescope-optics.net/
     
  10. Nov 16, 2014 #9
    Giving answers doesn't teach how to learn...
     
  11. Nov 16, 2014 #10

    mfb

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    Sure it does, just not in the lab. For very long distances it follows (to a very good approximation) the inverse square law as focussing cannot be perfect.
     
  12. Nov 16, 2014 #11

    Drakkith

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    If it doesn't follow it in the lab, and only begins to approximate it at large distances, that seems to imply that it doesn't follow the inverse square law.
     
  13. Nov 16, 2014 #12

    Drakkith

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    We're not here to teach people how to learn. We're here to teach people about physics.
     
  14. Nov 16, 2014 #13
    Then point the OP to a link on QED and let him "figure it out".
     
  15. Nov 16, 2014 #14

    Drakkith

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    Why would we do that? This is about optics, not QED.
     
  16. Nov 16, 2014 #15
    Since gravity has very little effect on optics, this should be covered far more precisely by quantum mechanics.
     
  17. Nov 16, 2014 #16
    And the OP wasn't here to "learn physics", the OP was here to understand the results of the experiment that was performed.
     
  18. Nov 16, 2014 #17

    Drakkith

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    Sure, if we were wondering about the quantum description of light and its interaction with the matter of the lens. But we aren't. That's far too in depth.

    Nonsense. The two are one and the same.

    It says nothing about the inverse square law, since it doesn't apply here. Also, remember that looking through a lens with your eye is not the same as taking that same lens and focusing light to a focal point with it. Looking through a lens at an object involves a complex optical system that uses multiple optical elements to form an image on your retina.

    Think of a "focal plane" instead of a "focal point". The image of whatever you are looking at is focused along a 2d plane known as the focal plane. The image formed at the focal plane of a lens with a short focal length is smaller than the image formed by a lens with a longer focal length. See the following images:

    u14l5c3.gif

    Note how the light from each point on the arrow is focused to a different point at the focal plane. While this picture only shows 3 points, there are actually an infinite number of them in every image. The light a the focal plane is the combined light from every point. In the next picture, the light rays are emanating from P1 in the above image. The light rays from P2 and P3 are not shown.

    Optics.jpg

    See how the shorter focal length lens at the top forms a smaller image than the longer focal length lens at the bottom? If we imagine the light from P2 and P3 being placed in this image, you can see that the points are spread further out in the bottom image. Since every image is composed of an infinite number of points, spreading them out means that we are also spreading the light out and the image formed is dimmer. The total amount of light is gathered by the lens and deposited at the focal plane for both cases. The only difference is that the amount of light deposited per unit of surface area is different, being higher for the top case and lower for the bottom case.
     
  19. Nov 16, 2014 #18
    This problem is better dealt with using geometrical optics not quantum mechanics. Indeed how would you use quantum mechanics?
    The system being investigated is not a spherical lens, it's a cylindrical lens and as such a good starting point would be to research the theory behind such lenses.
     
  20. Nov 16, 2014 #19

    mfb

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    The same is true for the light bulb as long as your distance to it is not significantly larger than the light bulb. It is just a quantitative difference, not a qualitative one.
     
  21. Nov 16, 2014 #20

    Drakkith

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    How so?
     
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