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			<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
			$$\left\{\begin{matrix}
			\Phi=L i \\
			\Phi=NBS\cos\alpha
		@@ -100,7 +100,7 @@
			\end{matrix}\right.$$
			Equating the two equations
			$$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$
		−	$$L=\mu_0 \frac{N^2}{l}S\cos^2\alpha$$
		+	$$L=\mu_0 \frac{N^2}{l}S\cos^2 \alpha$$
			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;">
			Lutfulloyev Shukurullo<br>
			Physics Department<br>
			National University of Uzbekistan<br>
			</p>


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	<header style="text-align:center;">		<header style="text-align:center;">
	<h2>Solutions of Savchenko Problems in Physics</h2>		<h2>Solutions of Savchenko Problems in Physics</h2>
	<p class="author">		<p class="author">
	Aliaksandr Melnichenka <br/>		Aliaksandr Melnichenka <br/>
	October 2023		October 2023
	</p>		</p>
	</header>		</header>

	<h3 id="back-link"><a href="../../#11.5">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#11.5">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<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 $N$.		$11.5.11^*.$ 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>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="statement.png"		<img src="statement.png"
	loading="lazy" width="200" />		loading="lazy" width="200" />
	<figcaption>		<figcaption>
	For problem $11.5.11^*$		For problem $11.5.11^*$
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	</p>		</p>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>

	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="draw1.png"		<img src="draw1.png"
	loading="lazy" width="280" />		loading="lazy" width="280" />
	<figcaption>		<figcaption>
	Direction of currents inside the solenoid		Direction of currents inside the solenoid
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	The following expressions follow from the figure		The following expressions follow from the figure
	$$i_\perp=i\cos\alpha$$		$$i_\perp=i\cos\alpha$$
	$$i_\parallel=i\sin\alpha$$		$$i_\parallel=i\sin\alpha$$
	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="draw2.png"		<img src="draw2.png"
	loading="lazy" width="280" />		loading="lazy" width="280" />
	<figcaption>		<figcaption>
	Geometry of the solenoid		Geometry of the solenoid
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	From the geometry of the solenoid, follows the aspect ratio		From the geometry of the solenoid, follows the aspect ratio
	$$\sin\alpha = \frac{r}{R}$$		$$\sin\alpha = \frac{r}{R}$$
	$$\cos\alpha = \sqrt{1-\frac{r^2}{R^2}}$$		$$\cos\alpha = \sqrt{1-\frac{r^2}{R^2}}$$
	Ampère's circuital law		Ampère's circuital law
	$$\oint B\,dl=\mu_0\sum_{k=1}^{N}i_k\Rightarrow B\cdot l=\mu_0 i_\perp N$$		$$\oint B\,dl=\mu_0\sum_{k=1}^{N}i_k\Rightarrow B\cdot l=\mu_0 i_\perp N$$
	By definition of the magnetic flux		By definition of the magnetic flux
	$$\left\{\begin{matrix}		$$\left\{\begin{matrix}
	\Phi=L i \\		\Phi=L i \\
	\Phi=NBS\cos\alpha		\Phi=NBS\cos\alpha
	@@ -100,7 +100,7 @@
	\end{matrix}\right.$$		\end{matrix}\right.$$
	Equating the two equations		Equating the two equations
	$$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$		$$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$
	$$L=\mu_0 \frac{N^2}{l}S\cos^2\alpha$$		$$L=\mu_0 \frac{N^2}{l}S\cos^2 \alpha$$
	Given the formula for the area of a circle $S=\pi r^2$, we substitute the value of $\cos\alpha$		Given the formula for the area of a circle $S=\pi r^2$, we substitute the value of $\cos\alpha$
	$$\boxed{L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}}$$		$$\boxed{L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}}$$
	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}$$		$$L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}$$
	</p>		</p>
	<p style="text-align: right; font-style: italic; font-size: 14;">		<p style="text-align: right; font-style: italic; font-size: 14;">
	Lutfulloyev Shukurullo<br>		Lutfulloyev Shukurullo<br>
	Physics Department<br>		Physics Department<br>
	National University of Uzbekistan<br>		National University of Uzbekistan<br>
	</p>		</p>


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