Dispersion and the dependences of refractive indexes

[Issy]

I'm revising for a uni exam with past exam papers, and have gotten stuck on the details of dispersion. The two exam questions prompting this are a) What is the physical reason why the index of refraction for blue light is bigger than that of red light? and b) Explain how dispersion makes a single lens give a blurry image.

I know that dispersion of light links the index of refraction of a medium to the frequency and/or wavelength of the light passing through it.
I'm stuck with the and/or part; when I'm answering questions like the ones above, (especially for a)), do I explain how different colours of light have different frequencies, or different wavelengths (even though both options are true)? I know that the frequency of light doesn't change, no matter the medium the light travels through, but the wavelength of the light does change, depending on medium. So would that mean (as the actual velocity of light appears to change when the light enters a different medium) that its index of refraction would really be dependent on wavelength rather than frequency?
And finally, regarding chromatic aberration, why is focal length of a lens dependent on refraction? Is there a (simple-ish) formula showing this? This is something I've just thought about as I was typing, so it's slightly unrelated, but I figured this was the right place to ask.

Relevant equations are n = c/v and v = fλ, and my attempts at solutions are
a) Dispersion - because as the frequency of blue light is higher than that of red light, no matter the wavelengths of each of them.
and
b) Lenses have different refractive indexes for different frequencies of light, which results in chromatic aberration as the focal length of a lens is dependent on its refractive index, which means that the different indexes of refraction for different colours of light will result in the different colours not focusing at the same point, thus creating a blurred image.

In typing out this question, things have begun to fall into place, but further information and/or insight would be welcomed. Thank you very much for any and all help!

 

[blue_leaf77]

I know that the frequency of light doesn't change, no matter the medium the light travels through, but the wavelength of the light does change, depending on medium. So would that mean (as the actual velocity of light appears to change when the light enters a different medium) that its index of refraction would really be dependent on wavelength rather than frequency?

The refractive index of a material can regarded as the response of that material into an incoming radiation as refractive index is a function of the dielectric constant of the material. Thus, the refractive index IS a function of frequency, but we can always change the dependency to the vacuum wavelength since we have the relation connecting vacuum wavelength and frequency. When a beam of light propagates through a medium, we know that its frequency remains unchanged (so long as we can assume the medium is linear) due to the conservation of photon energy, while the wavelength becomes different due to the slowing down of its phase velocity in the medium. Hence the change in the wavelength is caused by the change in the refractive index as the light travels from air into another medium, not the other way around as you indicated in your conclusion.

regarding chromatic aberration, why is focal length of a lens dependent on refraction? Is there a (simple-ish) formula showing this?

Have you heard about the lens maker formula?

a) Dispersion - because as the frequency of blue light is higher than that of red light, no matter the wavelengths of each of them.
and

The wavelength/frequency dependency of refractive index is what is called as dispersion. So you were actually only mentioning the synonym of the phenomena being asked. I would say the answer should indicate the reason why there is a rise in the refractive index as we go to higher frequency.
You probably need to know that the the so-called optical transitions is directly related to the absorption spectrum of a medium, and by virtue of Kramers-Kronig relation, the absorption and refractive index mutually influences each other - as you approach the absorption peak from lower frequency wing, up to certain point you will observe increasing refractive index (normal dispersion). In materials, we can classify the transition into electronics, vibrational, and rotational transitions which goes in the order of decreasing frequency in which each is typically observed. The electronics transition occurs typically in the UV and beyond frequency region, while the vibrational transitions are usually observed in infrared and lower. Therefore if we go to the higher frequency within the visible range we should indeed observe an increasing refractive index, because there are in most cases, absorption lines in the UV frequency region and beyond.