<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta property="og:title" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<title>What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.</title>
What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing $k$ times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is $n$ and the amount of plates is $N$.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" alt="13.2.11" width="300" />
<figcaption>
For problem 13.2.11
</figcaption>
</figure>
</center>
<h3>Solution</h3>
<p>
Let's consider the following figure
</p>
<center>
<figure>
<img src="draw.png"
loading="lazy" alt="13.2.11" width="450" />
<figcaption>
Ray's path through plates
</figcaption>
</figure>
</center>
<p>
Applying Snell's law for each interphase between media
$$n_0 \sin{\alpha} = n_1 \sin{\alpha_1} = n \sin{\alpha_1}$$
It's known that $n_2 = \frac{n}{k}$, $n_3 = \frac{n_2}{k}=\frac{n}{k^2}$, and so on. Then, $n_N=\frac{n}{k^{N-1}}$. The $N$-th refraction angle is $\frac{\pi}{2}$ beacause the ray doesn't pass this latest plate, i.e., there is a total reflection.
$$n_m \sin{\alpha} = \frac{n}{k^{N-1}}$$
If exterior medium is air, $n_m = 1$,
$$\boxed{\sin{\alpha}=\frac{n}{k^{N-1}}}$$
This angle $\alpha$ is the critical angle of the internal total reflection for this group of plates.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta name="description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta property="og:title" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta property="og:title" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta property="og:image" content="img/logo.png">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<meta property="og:description" content="What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.">
<title>What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.</title>
<title>What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing k times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is n and the amount of plates is N.</title>
What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing $k$ times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is $n$ and the amount of plates is $N$.
What minimum angle of incidence should have a light ray that incides over a group of plane transparent plates (each one with refraction index decreasing $k$ times respect to the upper to it) such that the ray doesn't pass the group? The refraction index of the upper plate is $n$ and the amount of plates is $N$.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" alt="13.2.11" width="300" />
loading="lazy" alt="13.2.11" width="300" />
<figcaption>
<figcaption>
For problem 13.2.11
For problem 13.2.11
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let's consider the following figure
Let's consider the following figure
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="draw.png"
<img src="draw.png"
loading="lazy" alt="13.2.11" width="450" />
loading="lazy" alt="13.2.11" width="450" />
<figcaption>
<figcaption>
Ray's path through plates
Ray's path through plates
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
Applying Snell's law for each interphase between media
Applying Snell's law for each interphase between media
$$n_0 \sin{\alpha} = n_1 \sin{\alpha_1} = n \sin{\alpha_1}$$
$$n_0 \sin{\alpha} = n_1 \sin{\alpha_1} = n \sin{\alpha_1}$$
It's known that $n_2 = \frac{n}{k}$, $n_3 = \frac{n_2}{k}=\frac{n}{k^2}$, and so on. Then, $n_N=\frac{n}{k^{N-1}}$. The $N$-th refraction angle is $\frac{\pi}{2}$ beacause the ray doesn't pass this latest plate, i.e., there is a total reflection.
It's known that $n_2 = \frac{n}{k}$, $n_3 = \frac{n_2}{k}=\frac{n}{k^2}$, and so on. Then, $n_N=\frac{n}{k^{N-1}}$. The $N$-th refraction angle is $\frac{\pi}{2}$ beacause the ray doesn't pass this latest plate, i.e., there is a total reflection.
$$n_m \sin{\alpha} = \frac{n}{k^{N-1}}$$
$$n_m \sin{\alpha} = \frac{n}{k^{N-1}}$$
If exterior medium is air, $n_m = 1$,
If exterior medium is air, $n_m = 1$,
$$\boxed{\sin{\alpha}=\frac{n}{k^{N-1}}}$$
$$\boxed{\sin{\alpha}=\frac{n}{k^{N-1}}}$$
This angle $\alpha$ is the critical angle of the internal total reflection for this group of plates.
This angle $\alpha$ is the critical angle of the internal total reflection for this group of plates.
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
