How Do You Calculate Group and Phase Velocities for a 550nm Wavelength?

[Dassinia]

Hello

Homework Statement

I have this tab
http://img18.imageshack.us/img18/3317/eyph.png

Anf I have to find the group and phase velocities for a wavelength λ=550nm

Homework Equations

v(phase)= c/n

v(group)=dw/dk

v(phase)*v(group)=c²/n²

The Attempt at a Solution

I don't know how I can determine n for the wavelength of 550nm using the tab ?
I was absent the day of this course
Can I use the relation v(phase)*v(group)=c²/n² ?

Thanks !

 

[mfb]

v(phase)= c/n
v(phase)*v(group)=c²/n²

Where do those two formulas come from? Where are they valid? Can you use them here?

I don't know how I can determine n for the wavelength of 550nm using the tab ?

Do an interpolation.

I was absent the day of this course

Interpolation is a much more general concept you should know.

 

[Dassinia]

The formulas come from my book, for electromagnetic waves.
Why can't I use them ?
OK, so I'll do an interpolation !
Thanks

 

[mfb]

Those equations would give v(phase)=v(group) everywhere, which is not true.
Please check how they got defined, and where they are applicable.

 

[Nickie]

Dear student,

Thank you for your question. Group and phase velocities are important concepts in the study of wave propagation. Group velocity refers to the speed at which the "envelope" or "group" of a wave packet travels, while phase velocity refers to the speed at which the individual wave crests travel. In other words, group velocity describes the overall propagation of a wave, while phase velocity describes the propagation of the individual components of the wave.

To determine the group and phase velocities for a given wavelength of 550nm, we can use the tab provided. The tab shows the refractive index (n) for different wavelengths (λ) in a medium. The refractive index is a measure of how much a wave is slowed down when traveling through a medium compared to its speed in a vacuum. We can use the equation v(phase)= c/n to calculate the phase velocity, where c is the speed of light in a vacuum (3x10^8 m/s). In this case, n for a wavelength of 550nm would be 1.5 (from the tab). Therefore, the phase velocity would be 2x10^8 m/s.

To calculate the group velocity, we can use the equation v(group)=dw/dk, where w is the angular frequency and k is the wave number. The angular frequency can be calculated using the equation w=2πf, where f is the frequency. The frequency can be calculated using the equation f=c/λ. Therefore, w=2πc/λ. The wave number can be calculated using the equation k=2π/λ. Therefore, v(group)=dw/dk= (2πc/λ)/(2π/λ)= c. This means that the group velocity is equal to the speed of light in a vacuum, which makes sense since the group velocity represents the overall propagation of the wave.

Alternatively, you can also use the relation v(phase)*v(group)=c²/n² to calculate the group velocity. Using the values we calculated before, we get v(group)= c²/(n*v(phase))= (3x10^8 m/s)²/(1.5*2x10^8 m/s)= 2x10^8 m/s. This matches the value we calculated using the previous method.

I hope this helps clarify the concepts of group and phase velocities for you. If you have any further questions, please do not hesitate to ask. Good luck