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<h2>Solutions of Savchenko Physics Textbook</h2>
<p class="author">
Aliaksandr Melnichenka <br/>
October 2023
</p>
</header>
− <main>
− <article>
− <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
−
− <h4>Statement</h4>
− <p>
− $13.2.11$
− 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>
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $13.2.11$
+ 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="statement.png"
− loading="lazy" alt="13.2.11" width="300" />
− <figcaption>
− For problem 13.2.11
− </figcaption>
− </figure>
− </center>
− <h4>Solution</h4>
− <p class="TxtSolutions">
− 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 class ="TxtSolutions">
− Applying Snell's law for each interphase between media
− </p>
− <center>
− $n_0 \sin{\alpha} = n_1 \sin{\alpha_1} = n \sin{\alpha_1}$
− </center>
− <center>
− $n \sin{\alpha_1} = n_2 \sin{\alpha_2}$
− </center>
− <center>
− $n_2 \sin{\alpha_2} = n_3 \sin{\alpha_3}$
− </center>
− <center>
− $\vdots$
− </center><center>
− $n_{N-1} \sin{\alpha_{N-1}} = n_N \sin{\frac{\pi}{2}} = n_N$
− </center>
− <p calss="TxtSolutions">
− 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.
− </p>
− <center>
− $n_m \sin{\alpha} = \frac{n}{k^{N-1}}$
− </center>
− <p class="TxtSolutions">
− If exterior medium is air, $n_m = 1$,
− </p>
− <center>
− $\boxed{\sin{\alpha}=\frac{n}{k^{N-1}}}$
− </center>
− <p class="TxtSolutions">
− This angle $\alpha$ is the critical angle of the internal total reflection for this group of plates.
− </p>
− <p class="TxtSolutions" style="text-align: right; font-style: italic; font-size: 14;">
− BSc. Luis Daniel Fernández Quintana<br>
− Physics Department (FCNE)<br>
− Universidad de Oriente, Cuba<br>
+ </center>
+ <p>
+ Applying Snell's law for each interphase between media
+ $$n_0 \sin{\alpha} = n_1 \sin{\alpha_1} = n \sin{\alpha_1}$$
+ $$n \sin{\alpha_1} = n_2 \sin{\alpha_2}$$
+ $$n_2 \sin{\alpha_2} = n_3 \sin{\alpha_3}$$
+ $$\vdots$$
+ $$n_{N-1} \sin{\alpha_{N-1}} = n_N \sin{\frac{\pi}{2}} = n_N$$
+ 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.
</p>
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+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ BSc. Luis Daniel Fernández Quintana<br>
+ Physics Department (FCNE)<br>
+ Universidad de Oriente, Cuba<br>
+ </p>
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