What is the general dependence of the phase velocity on wavelength in glass?

AI Thread Summary
The discussion focuses on the dependence of phase velocity (vp) on wavelength (λ) in glass, highlighting that glass is a dispersive medium. The phase velocity is expressed as vp = c/n, where n is the refractive index, which varies with wavelength. The participant is uncertain about how to derive vp(λ) and whether the velocity in the equation n = c/v refers to vp or another velocity. They propose that the derivative dvp/dλ is positive, indicating that phase velocity increases with wavelength. The conversation emphasizes the need for clarity in differentiating the various velocities involved in the equations.
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Homework Statement


a) Starting from Equation 5-1, show that the group velocity can also be expressed as: vg = vp - λ(dvp/dλ)

b) The phase velocity of each wavelength of white light moving through ordinary glass is a function of the wavelength; that is, glass is a dispersive medium. What is the general dependence of vp on λ in glass? Is dvp/dλ positive or negative?

I'm mostly concerned about part b, but I have a question about part a as well.

Homework Equations


Equation 5-1: f = E/h
vp = ω/k = fλ
vg = vp + k(dvp/dk)
n = c/v

3. The Attempt at a Solution

a) Using the last two equations I listed and just plugging in what k equals as well as what dk equals after differentiating the second equation in terms of wavelength, it is very easy to see how to get the the equation they want from the equation we were given in class. What I don't understand is how Eq. 5-1 comes into it, and what they want you to show "starting from" that equation. Any ideas?

b) I feel like this is simple but I'm not sure how to start because I am confusing all the different velocities and terms. Once I have vp(λ) I'll differentiate it to easily answer the second part. Any hints on how to begin to find vp(λ)? Is the v in n = c/v equal to vp, or is it equal to the actual velocity of the particle which is 2vp?
 
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anyone?

This is what I came up with for b:

vp = c/n
c = fλ

vp = fλ/n

dvp/dλ = f/n = positive

I have no idea if this is right. Seems too simple.
 
Last edited:
it seems right. Have you got your answer yet?
 

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