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## Homework Statement

I'm given the refractive index of a piece of glass:

$$n(\omega)=A+B\omega$$

And I have to find the speed at which a pulse of radiation will travel through the glass at an angular frequency $$\omega = 1.2 \times 10^{15} s^{-1}$$

I also have A = 1.4, B=3.00 x 10^-17.

## Homework Equations

I know that the pulse speed should be given by group speed, where

$$v_{group}=\frac{d\omega}{dk}$$

as opposed to

$$v_{phase}=\omega / k$$

## The Attempt at a Solution

$$k = \frac{n(\omega)\omega}{c}$$

$$v_{group}^{-1}=\frac{dk}{d\omega} = \frac{n(\omega)+n^{'}(\omega)\omega}{c}$$

$$v_{group}=\frac{c}{(A+B\omega)+B\omega}=\frac{c}{A+2B\omega}$$

Then I just put the figures in and get an answer. However, I have checked this answer by considering:

$$v_{phase} \times v_{group} = c^2$$

Using the formula for $$v_{phase}$$ above, and I don't seem to get the right answer. Is this identity above not always true?

Thanks in advance

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