Group velocity and the dispersion relation

[Kaguro]

Homework Statement

Q.Light of wavelength λ (in free space) propagates through a dispersive medium with refractive index n(λ)=1.5 + 0.6λ. The group velocity of a wave travelling inside this medium in units of 10^8 m/s is
(A) 1.5 (B) 2.0 (C) 3.0 (D)4.0

Relevant Equations

Group velocity is dw/dk. And phase velocity v is w/k.

After noting w=vk and differentiating with respect to k, and lots of simplifying, I get:

Vg = c/n +(2*pi*0.6)/(k*n)

This doesn't correspond to any numerical value though...

 

[kuruman]

You differentiated what with respect to k?

 

[Kaguro]

You differentiated what with respect to k?

I differentiated both sides of the equation w=vk w.r.t k

This will give me
dw/dk=v + k(dv/dk)

Group velocity is then dw/dk.
v is c/n the phase velocity..

and then I wrote k=2*pi/lamda
And substituted value of dk.

 

[kuruman]

It looks like you did not apply the chain rule correctly.

 

[Kaguro]

It looks like you did not apply the chain rule correctly.

Oh! That's right...

Now the value I get for group velocity is:

Vg=(c/n)+0.6*c*lamda/n^2

Is this correct?

Even so, this is not a numerical answer..

 

[kuruman]

Oh! That's right...

Now the value I get for group velocity is:

Vg=(c/n)+0.6*c*lamda/n^2

Is this correct?

Even so, this is not a numerical answer..

Look at your expression. It says that group velocity is equal to phase velocity plus something greater than zero. Can the group velocity be greater than the phase velocity? How did you get your answer. Please post the details of your work.

 

[Kaguro]

This is my work. Sorry for not typing all this out. That would have taken me hours..

 

[kuruman]

The photo is upside down. Next time please post any photos right side up. It's the considerate thing to do.

Your final expression looks algebraically correct.

 

[haruspex]

Oh! That's right...

Now the value I get for group velocity is:

Vg=(c/n)+0.6*c*lamda/n^2

Is this correct?

Even so, this is not a numerical answer..

Does this help https://en.m.wikipedia.org/wiki/Pha...ity,_refractive_index_and_transmission_speed?

 

[Replusz]

A quick check is to remember that group velocity is related to information transfer, hence it is always less than or equal to the speed of light in vacuum. Phase velocity, on the other hand, does not convey any information and can be greater than c.