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Homework Help: Maxwell's Equations - Wavelength Dependance derivation from Group Velocity

  1. Jan 21, 2010 #1
    1. The problem statement, all variables and given/known data

    For a particular Dieletric it is observed that over a range of frequencies, the group velocity varies exponetinally with wave number:

    v[tex]_{g}[/tex]=ae[tex]^{bk}[/tex] , where a,b are constants. *(PLease not that the superscript g on v on the LHS side is meant to be subscript, however it wont let me change it....)*

    Derive the wavelength dependance of the refractive index for this material.


    2. Relevant equations

    Maxwell's equations in dieletric media.......?


    3. The attempt at a solution

    This is a revision question for an upcoming exam. I dont really have any idea at how to approach this question. The only thing that comes to mind is that the refractive index is the ratio of the velocity in two media. I think this is quite an easy question but dont know how to do it! Any help or guidance would be greatly appreciated.

    Thanks, Leo
     
  2. jcsd
  3. Jan 21, 2010 #2
    The definition of refractive index is

    [tex]
    n = \frac{c}{v_{phase}}.
    [/tex] ----- (1)

    The definitions of phase and group velocities are

    [tex]
    v_{phase} = \frac{\omega}{k};
    [/tex] ----- (2)

    [tex]
    v_{group} = \frac{\partial\omega}{\partial k}.
    [/tex] ----- (3)

    Since the [tex]v_{group}(k)[/tex] function is given we can derive the [tex]\omega(k)[/tex] function, using (3):

    [tex]
    \omega(k) = \frac{a}{b}e^{bk} + d,
    [/tex] ----- (4)

    where [tex]d[/tex] is a constant.
    (2) and (4) will give us:

    [tex]
    v_{phase} = \frac{a}{bk}e^{bk} + \frac{d}{k},
    [/tex]

    or

    [tex]
    n(k) = \frac{kc}{(a/b)\exp(bk) + d}.
    [/tex]

    It's known that

    [tex]
    k = \frac{2\pi}{\lambda}
    [/tex]

    so the wavelength dependence will be

    [tex]
    n(\lambda) = \frac{c}{\lambda[(a/b)\exp(b/\lambda) + d]}.
    [/tex]
     
  4. Jan 21, 2010 #3
    Cheers buddy that makes sense, quitre simple really just didn't think about w(k). Thanks for your help!
     
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