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Absorption of Light

  1. Oct 18, 2015 #1
    1. The problem statement, all variables and given/known data
    A slab of glass 0.5 m thick absorbs 50% of light passing through it. Compute how thick of a slab of identical glass you would need that:
    - absorbs 90% of light
    - absorbs 99% of light
    - absorbs 99.9% of light

    2. Relevant equations
    I spent an hour looking for formulas in my text book and couldn't find any, which is the primary reason I'm having trouble.

    3. The attempt at a solution
    N/A
     
  2. jcsd
  3. Oct 18, 2015 #2

    blue_leaf77

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    The simplest mathematical formulation to describe light absorption is through the equation ##I(z) = e^{-\alpha z} I(0)##, where ##z## the distance traveled and ##\alpha## an absorption coefficient.
     
  4. Oct 18, 2015 #3
    I typed light absorption as a function of glass thickness into the Google search engine and it came up on the first hit.
     
  5. Oct 18, 2015 #4
    So the quantity that we are trying to measure here is intensity? I do see a formula in my textbook saying I(z) = I0e-τ(x). Could that be it?

    EDIT: This actually appears to be a formula for intensity through a medium of gas. Nevermind.
     
  6. Oct 18, 2015 #5

    blue_leaf77

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    Shouldn't the exponent be a function of z?
     
  7. Oct 18, 2015 #6
    Yes, I mistyped it. Both I and tau should be functions of x according to my textbook. I'm still not finding a formula for absorption of light by a solid anywhere in my textbook, only for gases.
     
  8. Oct 18, 2015 #7

    blue_leaf77

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    If you can't find it in your book, you should look for it somewhere else, shouldn't you?
     
  9. Oct 18, 2015 #8
    Another formula in my textbook says ΔI/Δ = -(nSΔx)σ/S = -nσΔx. σ here is the cross-section, Δx the thickness, and n the number density. For the sake of this problem, the number density and cross-section would remain constant with Δx as the only independent variable to change ΔI/I. Would this be an acceptable solution to this problem?
     
  10. Oct 18, 2015 #9

    blue_leaf77

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    This is accurate only when the thickness and/or the absorption coefficient are small. Look what happen if you make the deltas infinitesimal (##\Delta \rightarrow d##),
    $$
    \frac{dI}{I} = n\sigma dx
    $$
    and then integrate both sides. What expression will you get?
     
  11. Oct 18, 2015 #10
    What would two of these slabs do?

    Perhaps you underestimate your own powers of reason.
     
  12. Oct 18, 2015 #11
    lnI = -nσ dx + C1.
    If I use the knowledge that at x = 0 I = I0 we can use differential equations (I worked them out on a sheet of paper) to get C1 = lnI0 and then I(x) = I0e-nσx.

    I think...
     
  13. Oct 18, 2015 #12

    blue_leaf77

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    Yes, that's right, and how does it compare to the equation in comment #2?
     
  14. Oct 18, 2015 #13
    It's looks similar. In the equation in comment 2 I would assume number density and cross section were combined into a single constant.
     
  15. Oct 18, 2015 #14

    blue_leaf77

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    So, what is still halting you to answer your homework questions?
     
  16. Oct 18, 2015 #15
    Nothing anymore. Thank you very much!
     
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