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If there is a ray of light moving from $n_1$ to $n_2$ you can get dispersion if $n_1$ is a function of wavelength. What angle of incidence ($\theta_1$) will maximise the dispersion? enter image description here

My solution so far: Well I looked at Snell's law which is as follows $$ n_1sin(\theta_1) = n_2sin(\theta_2) $$ Now we know $n_2=1$ (given in question) and to find maximum dispersion we can say $\theta_2 = \frac{\pi}{2}$ and thus $sin(\theta_2) = 1$. Now our equation becomes $$ n_1sin(\theta_1) = 1$$ Now I get a little bit stuck, but what I thought is we know $n_1$ is a function of wavelength and we are trying to find a formula for the critical angle for any wavelength you give me (i.e. that if you give me any wavelength then I can give you a $n_1$ and hence tell you what your critical angle for Total Internal Reflection). I want to try and use the fact that $n=\frac{\lambda}{\lambda_m} $ but then I need to know what the wavelength of light is which can obviously be a range of values. Please can anyone help with this part or give me some tips on how to tackle it?

P.S. I apologise if i explained this poorly but if you are unsure on what I am asking this asking just say so and i will try to explain a different way.

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You're simply meant to solve your equation $ n_1sin(\theta_1) = 1$ for $\theta_1$ at one wavelength of light. $n_1$ is (assumedly) known at that wavelength, and $\theta_1$ is the unknown critical angle.

I don't think I've ever seen an undergraduate physics question - at least not about Snell's law, that asks a student to work with wavelength varying refractive index.

In fact, the refractive index's variation with wavelength is extremely material dependent and depends in detail on the molecular resonances in the molecules in question. Glassmakers dope their mixtures with this and that to tailor the refractive index with wavelength variation for different lens design functions. Look up the Sellmeier equations which are used by the big manufacturers like Schott and Ohara - they are square-root eighth order and higher rational functions and much more complicated than you allude to here.

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