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+ <meta name="author" content="Aliaksandr Melnichenka">
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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Problems in Physics</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../#11.5">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $11.5.11^*.$ Calculate the inductance of a long solenoid of radius $r$ and length $l$ placed inside a long superconducting tube of radius $R$ along its axis. Number of turns of the solenoid $N$.
+</p>
+<center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" width="200" />
+ <figcaption>
+ For problem $11.5.11^*$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+
+</p>
+<center>
+ <figure>
+ <img src="draw1.png"
+ loading="lazy" width="280" />
+ <figcaption>
+ Direction of currents inside the solenoid
+ </figcaption>
+ </figure>
+</center>
+<p>
+The following expressions follow from the figure
+$$i_\perp=i\cos\alpha$$
+$$i_\parallel=i\sin\alpha$$
+</p>
+<center>
+ <figure>
+ <img src="draw2.png"
+ loading="lazy" width="280" />
+ <figcaption>
+ Geometry of the solenoid
+ </figcaption>
+ </figure>
+</center>
+<p>
+From the geometry of the solenoid, follows the aspect ratio
+$$\sin\alpha = \frac{r}{R}$$
+$$\cos\alpha = \sqrt{1-\frac{r^2}{R^2}}$$
+Ampère's circuital law
+$$\oint B\,dl=\mu_0\sum_{k=1}^{N}i_k\Rightarrow B\cdot l=\mu_0 i_\perp N$$
+By definition of the magnetic flux
+$$\left\{\begin{matrix}
+\Phi=L i \\
+\Phi=NBS\cos\alpha
+\end{matrix}\right.$$
+Equating the two equations
+$$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$
+$$L=\mu_0 \frac{N^2}{l}S\cos^2\alpha$$
+Given the formula for the area of a circle $S=\pi r^2$, we substitute the value of $\cos\alpha$
+$$\boxed{L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}}$$
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}$$
+ </p>
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ Yuldashev Ulugbek<br>
+ </p>
+
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