Added 7.1.12, 7.1.23, 7.3.9 & 11.5.11

astrosander edited
View solution History
revision #9813 parent #4129 GitHub 5aa3af1 ← older newer →
en/11.5.11.md +128 −0
@@ -0,0 +1,128 @@
+<!DOCTYPE html>
+<html lang="en">
+
+<head>
+ <meta charset="utf-8">
+ <meta name="viewport" content="width=device-width, initial-scale=1.0">
+ <meta http-equiv="content-language" content="en">
+ <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
+ <meta name="description" content="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.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="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.">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="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.">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>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.</title>
+ <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
+ <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png">
+ <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script>
+ <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script>
+ <script type="text/x-mathjax-config">
+ MathJax.Hub.Config({
+ extensions: ['tex2jax.js'],
+ jax: ['input/TeX', 'output/HTML-CSS'],
+ tex2jax: {
+ inlineMath: [['$', '$'], ['$', '$']],
+ processEscapes: true,
+ processClass: 'tex2jax',
+ ignoreClass: 'html'
+ },
+ showProcessingMessages: false,
+ messageStyle: 'none'
+ });
+ </script>
+</head>
+ <body style="">
+ <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;">
+ Lutfulloyev Shukurullo<br>
+ </p>
+
+
+ <footer class="row container">
+ <br>
+ <p>
+ <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
+ </p>
+ <p>
+ <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
+ </p>
+ </footer>
+ </body>
+
+</html>