Light is incident at an angle alpha on one planar end of a transparent cylindrical rod of refractive

Q: Why do we take the maximum value of $\alpha$ as $90^\circ$?

The problem states the condition must hold *irrespective* of $\alpha$. To ensure the light never escapes, we must secure the worst-case scenario. The light is most likely to escape when it strikes the curved surface at the steepest angle, which happens when it enters the flat face at grazing incidence ($\alpha \to 90^\circ$).

Q: Why is the angle of incidence at the curved surface $90^\circ - r$?

If you draw a normal at the flat face (horizontal) and a normal at the curved surface (vertical), they intersect at a $90^\circ$ angle. The refracted ray forms a right-angled triangle with these normals, making the two acute angles complementary. Hence, $r' = 90^\circ - r$.

Q: What is the critical angle $\theta_c$?

The critical angle is the specific angle of incidence beyond which total internal reflection occurs. It is defined by $\sin \theta_c = \frac{n_{\text{outside}}}{n_{\text{inside}}}$. Since the outside is air ($n=1$), $\sin \theta_c = \frac{1}{n}$.

Q: Why did we flip the inequality when taking the sine?

We didn't flip it when taking the sine, because sine is an increasing function from $0^\circ$ to $90^\circ$. We flipped it at the very end when we took the reciprocal: if $\frac{1}{n} \le \frac{1}{\sqrt{2}}$, then $n \ge \sqrt{2}$.

Q: Does this principle have real-world applications?

Absolutely! This is the foundational physics behind optical fibers. By choosing a core material with a high enough refractive index relative to its cladding, engineers ensure that light signals injected into the fiber are perfectly trapped and can travel across oceans without leaking out.

Question

Light is incident at an angle $\alpha$ on one planar end of a transparent cylindrical rod of refractive index $n$. Determine the least value of $n$ so that the light entering the rod does not emerge from the curved surface of the rod irrespective of the value of $\alpha$.

Sigma Percentile · Accepted Answer

The journey of a light ray through a transparent medium is a beautiful dance of geometry and physics. In this problem, we are tasked with designing a cylindrical rod that acts as a perfect light trap. No matter how the light enters the flat end, it must never escape through the curved sides.

This is not just an abstract textbook problem; it is the exact engineering principle behind the optical fibers that power our global internet! Let's break down the physics of this perfect light trap.

### Analyzing the Setup

Imagine a light ray striking the flat planar end of the cylindrical rod. It arrives at an angle of incidence $\alpha$ relative to the central axis (which acts as the normal to the flat face).

As the light enters the denser medium of the rod (refractive index $n$), it bends towards the normal. Let's call this angle of refraction $r$. According to Snell's Law at this first interface:

$$ 1 \cdot \sin \alpha = n \sin r $$

Once inside, the ray travels towards the top curved surface. If we draw a normal to the curved surface at the point of impact, we can see that it forms a right-angled triangle with the central axis. Because the sum of angles in a triangle is $180^\circ$, the angle of incidence at the curved surface, let's call it $r'$, is simply the complement of $r$:

$$ r' = 90^\circ - r $$

### The Master Equation for Trapping Light

For the light to remain trapped inside the rod, it must undergo Total Internal Reflection (TIR) at the curved surface. The fundamental condition for TIR is that the angle of incidence must be greater than or equal to the critical angle $	heta_c$ of the material.

$$ r' \ge 	heta_c $$

We know that the critical angle is defined by the refractive index of the medium relative to the outside air:

$$ \sin 	heta_c = \frac{1}{n} $$

Here is the crucial part of the problem: the light must be trapped **irrespective of the value of $\alpha$**. This means our TIR condition must hold true even in the absolute worst-case scenario.

When is the light most likely to escape? It will escape if $r'$ is as small as possible. Since $r' = 90^\circ - r$, minimizing $r'$ means we must maximize $r$. And $r$ is maximized when the incoming angle $\alpha$ is at its absolute maximum.

The maximum possible angle for light entering the flat face is grazing incidence, where $\alpha 	o 90^\circ$. At this extreme limit, $\sin \alpha 	o 1$. Plugging this into Snell's Law:

$$ 1 = n \sin r_{\max} \implies \sin r_{\max} = \frac{1}{n} $$

Wait a minute! We just established that $\frac{1}{n}$ is exactly $\sin 	heta_c$. This leads to a beautiful geometric revelation:

$$ r_{\max} = 	heta_c $$

### Final Calculation

Now we know the worst-case scenario. The minimum possible angle of incidence at the curved surface is:

$$ r'_{\min} = 90^\circ - r_{\max} = 90^\circ - 	heta_c $$

For the rod to be a perfect light trap, even this minimum angle must be sufficient to trigger TIR:

$$ r'_{\min} \ge 	heta_c $$

Substituting our expression for $r'_{\min}$:

$$ 90^\circ - 	heta_c \ge 	heta_c $$

$$ 90^\circ \ge 2	heta_c \implies 	heta_c \le 45^\circ $$

To find the refractive index $n$, we take the sine of both sides. Since the sine function is strictly increasing between $0^\circ$ and $90^\circ$, the inequality direction remains the same:

$$ \sin 	heta_c \le \sin 45^\circ $$

$$ \frac{1}{n} \le \frac{1}{\sqrt{2}} $$

Flipping the fractions reverses the inequality:

$$ n \ge \sqrt{2} $$

And there we have it! To guarantee that light never escapes the sides of the rod, the material must have a refractive index of at least $\sqrt{2}$ (approximately $1.414$). This elegant mathematical result is the cornerstone of modern fiber optics!

Visualized Solution

\text{Visualizing the Setup}

\text{Snell's Law at the Planar Face}

\text{Geometry of the Ray}

\text{Condition for Total Internal Reflection}

\text{Worst-Case Scenario}

\text{Minimizing } r'

\text{Maximizing } \alpha

\text{Finding } r_{\max}

\text{Substituting } r_{\max}

\text{Applying the TIR Condition}

\text{Solving for } \theta_c

\text{Final Calculation}

\text{The Way Forward}

The Sigma Insight: Refraction and Total Internal Reflection

Analyzing the Setup

The Master Equation for Trapping Light

Final Calculation

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