# Can Snell's law be said to fail in this condition?

I ran into a question in my textbook which asks me "when does Snell's law fail?". My first reaction was that it's a law, it can't fail but they gave the solution as the condition when light rays are incident at the interface normally i.e. when $$\angle i=\angle r=0$$.

$$\angle i=\angle r=0 \implies \dfrac{\sin i}{\sin r} \ \text{is indeterminate}$$

Could someone explain this? I would not think it as being a failure of Snell's law. Can we somehow think of making use of Calculus and make use of the fact that both functions in the numerator and denominator are of the same order so the fraction must converge to some finite value? Also, if this indeed is a failure would we say the case when Total Internal Reflection happens, that is a failure too?

• This appears as a really weird statement to me. All you have to do is to re-write it as $n_isin(i)=n_rsin(r)$, and the law is clearly not "failing" as you just get 0=0. And total reflection can perfectly well be described in terms of Snell law (to compute the critical angle for example), so it is clearly not a failure either – Barbaud Julien Mar 12 '19 at 5:53
• Snell's law and total internal reflection – exp ikx Mar 12 '19 at 6:00
• quora.com/Where-does-Snells-law-of-refraction-fail – SRS Mar 12 '19 at 6:18
• @BarbaudJulien This is not a helpful way to think IMO. You can't just rearrange the terms and forget about the indeterminate quantities that otherwise arise. Indeterminacy means something. In this case, the indeterminacy corresponds to the fact that regardless of the ratio of the refractive indices of the two media, a ray incident normally would transmit normally. – Dvij Mankad Mar 14 '19 at 4:07

Snell's law is a law that relates the incident angle $$i$$ and the refraction angle $$r$$ for an incident medium with a refractive index $$n_i$$ and a refractive medium with a refractive index $$n_r$$. In particular, $$n_i\sin i=n_r \sin r$$. Thus, I will treat it as a law pertaining to the phenomenon of refraction.
The law is, thus, moot if there is no refracted ray, i.e., if the ray gets internally reflected within the incident medium. Thus, the law is not violated--it is simply moot because the law never was about internally reflected rays. However, the flipside of Snell's law being the valid law for all the phenomena of refraction is that Snell's law puts a restriction on the incident angle for which it can get refracted into the refractive medium. Because, if Snell's law is really valid for all the phenomena of refraction, it means that there cannot be a phenomenon of refraction for $$i>\arcsin\Big(\frac{n_r}{n_i}\Big)$$. This restriction is, of course, correct.
In the case of both the incident and the refraction angle being zero, the law $$n_i\sin i=n_r \sin r$$ is not violated--assuming the refractive indices of both the media are finite. Both the sides of the equation assuredly vanish and the law holds. Now, if you want to view the law which gives the ratio of the refractive indices of the two media given the relevant angles, in the case of $$i=0$$, it still tells you that $$r$$ has to vanish if both the refractive indices are bound to be finite. If either of $$i$$ or $$r$$ vanishes but the other does not, the validity of Snell's law would require either of the refractive indices to diverge. Thus, the assumption of the finiteness of both of the refractive indices allows Snell's law to be valid as well as to predict that either of $$i$$ or $$r$$ can vanish only if the other vanishes too. However, all of this still doesn't allow you to know the ratio $$\frac{n_i}{n_r}$$ in the case of $$i=r=0$$. But this is not a violation of Snell's law because (again, assuming the finiteness of both the refractive indices) $$n_i\sin_i=n_r\sin_r$$ still holds. This simply tells you that no matter what combination of finite refractive indices one chooses for the two media, a ray which is incident normally will transit normally--which is, of course, correct.
This phenomenon was already explained by Christian Huygens in his Traité de la Lumière où sont expliquées les causes de ... l'étrange réfraction du cristal d'Islande, by assuming different speeds for different propagation direction in substances like calcite, so that wave fronts ar not spherical anymore but ellipsoidal around the optical axis. 