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

Click For Summary
SUMMARY

The discussion centers 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 c is the speed of light and n is the refractive index. The derivative dvp/dλ is determined to be positive, indicating that as wavelength increases, phase velocity also increases. The relationship is derived from the equations f = E/h, vp = fλ, and n = c/v.

PREREQUISITES
  • Understanding of wave mechanics and dispersion in optics.
  • Familiarity with the equations of motion for waves, particularly vp = fλ.
  • Knowledge of refractive index and its relationship to phase velocity.
  • Basic calculus for differentiation of functions related to wavelength.
NEXT STEPS
  • Study the concept of dispersion in optical materials, focusing on glass.
  • Learn about the derivation and implications of the equation vp = c/n.
  • Explore the relationship between group velocity (vg) and phase velocity (vp) in dispersive media.
  • Investigate the impact of wavelength on the refractive index of various materials.
USEFUL FOR

Students studying optics, physicists analyzing wave behavior in materials, and educators teaching concepts of wave mechanics and dispersion.

Kavorka
Messages
95
Reaction score
0

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?
 
Physics news on Phys.org
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?
 

Similar threads

What is the phase velocity of light in glass?
  • · Replies 2 ·
Replies
2
Views
2K
Why Does Light Diminish with a Phase Difference of 5λ?
  • · Replies 6 ·
Replies
6
Views
1K
What are the speeds of P and M while riding Ents and playing catch with rings?
  • · Replies 18 ·
Replies
18
Views
2K
Phase velocity of EM radiation in free space
  • · Replies 7 ·
Replies
7
Views
4K
Group and phase velocity of a wave
  • · Replies 2 ·
Replies
2
Views
2K
Total Internal Reflection and Transmitted Wavelength
  • · Replies 1 ·
Replies
1
Views
948
Phase difference after passing through liquid and glass
  • · Replies 2 ·
Replies
2
Views
2K
Doppler effect and acceleration of source
  • · Replies 5 ·
Replies
5
Views
2K
Phase difference between points on wall
  • · Replies 4 ·
Replies
4
Views
3K
What Visible Wavelengths Cause Maximum Constructive Interference in a Thin Film?
  • · Replies 1 ·
Replies
1
Views
3K