# Group Velocity of Waves in Gas Problem

## Homework Statement

The dielectric constant k of a gas is related to its index of refraction by the relation $k = n^{2}$.

a. Show that the group velocity for waves traveling in the gas may be expressed in terms of the dielectric constant by

$\frac{c}{\sqrt{k}}(1 - \frac{ω}{2k}\frac{dk}{dω}$

where c is the speed of light in vacuum.

## Homework Equations

$v_{g} = v_{p}(1 + \frac{λ}{n}\frac{dn}{dλ})$ (1)
$v_{p} = \frac{c}{n}$ (2)

## The Attempt at a Solution

Plugging (2) into one

$v_{g} = \frac{c}{n}(1 + \frac{λ}{n}\frac{dn}{dλ})$ (3)

Taking the given information and solving for n

$k = n^{2}, n = \sqrt{k}$

Plugging this into (3)

$v_{g} = \frac{c}{sqrt(k)}(1 + \frac{λ}{sqrt(k)}\frac{dn}{dλ})$

I'm not really sure where to go from here. I would imagine I need to some how find λ as a function of n and take the derivative of this function. I would imagine that this function is also a function of ω and k in some way. I'm not sure of what this equation is though. I have looked through my book in the chapter in which this problem was given and can find no equation.

Thanks for any help.

Related Introductory Physics Homework Help News on Phys.org
This is a chain rule exercise.

I'm not really sure how because I don't have an equation for ω and am not really sure how it related to the problem. Is it rotational speed in this case or another variable? I don't see how rotational speed relates to k in this case and I believe is where I'm getting stuck.

ω is the angular frequency.
ω=2∏f where f is the frequency in Hz.
Now you should know a relationship between λ and ω.

I'm not really sure how because I don't have an equation for ω and am not really sure how it related to the problem. Is it rotational speed in this case or another variable? I don't see how rotational speed relates to k in this case and I believe is where I'm getting stuck.
Even though nasu explained how you should proceed, I would like to point out that you should have said that you do not understand the meaning of $omega$ in the original post. Obviously, if one does not even understand the specification of the problem, trying anything could solve it only by chance (unlikely).