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Wavelength and resolution

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data
    the wavelength of the probing beam must be approximately the same as the size of objects we want to resolve. For each of the accelerators described below, determine the smallest size objects that are resolvable and name an object roughly that small, remembering that these are order-of-magnitude estimates.
    (a) In the radioactive decay of radium, most emitted [tex]\alpha[/tex]particles (4He nuclei) travel at 1.5 ×10^7 m/s. This is what was used in the Geiger-Marsden Experiment.
    (there are actually four questions, but I am trying to make sure I am on the right track)
    2. Relevant equations
    p=[tex]\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/tex]

    p=[tex]\frac{h}{\lambda}[/tex]

    [tex]\lambda[/tex]=[tex]\frac{h}{p}[/tex]

    where h = Planck's constant

    3. The attempt at a solution

    Sorry, I am not too good with LATEX references...

    a) v= 1.5e7 m/s
    m=6.64e-27 kg

    p=[tex]\frac{(6.64e-27 kg)(1.5e7 m/s)}{\sqrt{1-\frac{(1.5e7 m/s)^{2}}{3e8m/s^{2}}}}
    [/tex]

    I solve this equation to get momentum 9.97e20 kg*m/s, then plug it into the next formula to find the wavelength:
    [tex]\lambda[/tex]=[tex]\frac{6.626 J*s}{9.97e20 kg*m/s}[/tex]

    I get this answer:

    [tex]\lambda[/tex]=6.65e-55

    which is much smaller than an electron, quark or a string.

    Surely I did something wrong, but I can't find my mistake.
     
  2. jcsd
  3. Nov 17, 2009 #2

    kuruman

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    Redo the momentum calculation. Your value of order 1020 kg m/s is absurdly large for an alpha particle.

    *** Additional comment on edit ***
    Also your value of h is missing the appropriate powers of 10.
     
    Last edited: Nov 17, 2009
  4. Nov 17, 2009 #3
    Thanks so much. I missed the math error, I didn't have the negative sign for the scientific notation.
    I now have the answer of 6.65e-15m, which is about the size (as far as orders of magnitude go) of an electron.
    That was the answer I was looking for!
    Thanks again
     
  5. Nov 17, 2009 #4

    ideasrule

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    Where does it say electrons are 10^-15 m across? Electrons, as far as we know, have 0 size. Experiments have certainly constrained its radius to no larger than 10^-18 m.
     
  6. Nov 17, 2009 #5
    Last edited: Nov 17, 2009
  7. Nov 18, 2009 #6

    Redbelly98

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    Note, that link actually says electrons are somewhere between 0 and 1e-15 meters.
     
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