# Is Snell’s Law valid in this case?

When light travels in a perpendicular path from one medium to another medium of different optical density, is Snell’s law valid?

$$\sin i$$ and $$\sin r$$ are both 0, right? So it isn’t valid.

Is this correct?

• How is it not valid? You get $0$ for each angle, which is precisely what happens. – BioPhysicist Jan 15 at 5:33
• Yes, but 0/0 has no defined value. But the refractive index of the second medium (say, glass) has a defined value. Then? – Dora Jan 15 at 5:46
• @Dora Snell's law is usually written $n_1\sin(\theta_1) = n_2\sin(\theta_2)$, no division involved. – J. Murray Jan 15 at 5:58
• @J. Murray, what if you write it as sin theta 1/ sin theta 2 = n2 / n1? – Dora Jan 15 at 9:11
• @Dora You obtain that form by dividing the expression I wrote by $\sin(\theta_2)$, which you cannot do if $\sin(\theta_2)=0$. – J. Murray Jan 15 at 13:26

In this case where the incident angle is $$0^\circ$$ to the interface and thus from $$n_{1}\sin{\theta_1}=n_2\sin{\theta_2}$$ we get that $$\theta_1 = \theta_2$$ I'd simply instead reason and say that Snell's law is not applicable in this case as it'd result in the forbidden $$\frac{0}{0}$$ formulation. Instead as Snell's law, if we consider these cases, it might be more valid to refer to it as Snell's model.

Let me first give an example from Newton's laws. If there is no net force acting on an object, then it will not be accelerating. Therefore, the equation of Newton's second law $$F=ma$$ is a valid equation, as we have $$0=m\cdot 0=0$$. But what if we were looking at this scenario with the equation $$F/a=m$$? This gives us $$0/0$$ on the left side of the equation, and it's no longer valid. What happened?

Mathematically, $$0/0$$ is undefined, so it was invalid to divide by $$a$$ in the first place. However, physically this means here our mass is undefined. But this makes sense. For any mass, $$0$$ net force means $$0$$ acceleration, so in this scenario with only this information we cannot determine the mass of the object. Therefore, it makes sense that we get an undefined mass.

Moving onto your example, mathematically it is invalid to do what you propose, as you are dividing by $$0$$. However physically, for any two adjacent media, a light ray incoming perpendicular to their interface will not refract. So just knowing that both angles are $$0$$ cannot give us any information about the indices of refraction of the two media. Therefore, it makes sense that we get an undefined value in this version of the equation.

In any case, just like in the Newton's law example, it is better to just keep Snell's law without the division in this case. Then you have a valid equation that reads $$0=0$$.