Translated 9.2.10-9.2.17

astrosander правка от 17 окт. 2024 г., 19:43

правка #11459 GitHub ef227c0 позже →

en/9.2.11.md +107 −0

		@@ -0,0 +1,107 @@
		+	<!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="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
		+	<meta name="author" content="Aliaksandr Melnichenka">
		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
		+	<meta property="og:title" content="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
		+	<meta property="og:image" content="img/logo.png">
		+	<meta property="og:description" content="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
		+	<meta name="yandex-verification" content="6cfda41f74038368">
		+	<title>How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.</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;">
		+	<a href="../../" style="text-decoration: none;">
		+	<div id="logo">
		+	<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
		+	</div>
		+	</a>
		+	<p class="author">
		+	Solutions of Savchenko Problems in Physics <br>
		+	<i><b>knowledge must be free</b></i>
		+	</p>
		+	</header>
		+
		+	<h3 id="back-link"><a href="../../#9.2">$\leftarrow$Back</a></h3>
		+
		+	<h3> Statement </h3>
		+	<p>
		+	$9.2.11.$ How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle $\alpha$? The current in the ring does not change.
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/9/9.2.11/9.2.11_1.png"
		+	loading="lazy" width="200" />
		+	<figcaption>
		+	Magnetic field induction at the center of a bent ring
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	In <a href="../9.2.10">9.2.10</a> we obtained an expression for the magnetic induction at the center of the ring
		+	$$ B_0 = \frac{\mu_0 I}{2R}\quad(1) $$
		+	Due to superposition, the magnetic induction at the center of one semicircle will be half of this value:
		+	$$ B_1 = B_2 = \frac{B_0}{2} = \frac{\mu_0 I}{4R}\quad(2) $$
		+	From the drawing, it is clear that the angle between the vectors $\vec{B}_1$ and $\vec{B}_2$ is equal to $\alpha$
		+	$$ (\vec{B_1},\vec{B_2})=\alpha $$
		+	The total vector of magnetic induction is equal to the vector sum of the components of magnetic induction of two rings $\vec{B}_1$ and $\vec{B}_2$
		+	$$ \vec{B} = \vec{B_1} + \vec{B_2} $$
		+	We will find the modulus of this vector using the cosine law.
		+	$$ B = \sqrt{{B_1}^2 + {B_2}^2 - 2B_1 B_2 \cos\alpha} $$
		+	Taking into account the expression $(2)$, we find the total magnetic induction as
		+	$$ B= \frac{\mu_0 I}{4R}\sqrt{2-2\cos\alpha} = \frac{\mu_0 I}{2R}\cdot\sin\frac{\alpha}{2}\quad(3) $$
		+	Dividing the expression $(3)$ by $(1)$, we obtain the desired ratio
		+	$$ \boxed{\frac{B}{B_0} = \sin\frac{\alpha}{2}} $$
		+
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$\frac{B}{B_0} =\sin\frac{\alpha}{2}$$
		+	</p>
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Andrei Yersh<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> alex@savchenkosolutions.com <br></small>
		+	</p>
		+	</footer>
		+	</body>
		+
		+	</html>

	@@ -0,0 +1,107 @@
			<!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="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle \alpha? The current in the ring does not change.</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;">
			<a href="../../" style="text-decoration: none;">
			<div id="logo">
			<span><img src="../../img/book.png"></span><span>Savchenko Solutions</span>
			</div>
			</a>
			<p class="author">
			Solutions of Savchenko Problems in Physics <br>
			<i><b>knowledge must be free</b></i>
			</p>
			</header>

			<h3 id="back-link"><a href="../../#9.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$9.2.11.$ How many times will the magnetic field induction in the center of a ring with current decrease if it is bent at an angle $\alpha$? The current in the ring does not change.
			</p>

			<h3>Solution</h3>
			<p>

			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/9/9.2.11/9.2.11_1.png"
			loading="lazy" width="200" />
			<figcaption>
			Magnetic field induction at the center of a bent ring
			</figcaption>
			</figure>
			</center>
			<p>
			In <a href="../9.2.10">9.2.10</a> we obtained an expression for the magnetic induction at the center of the ring
			$$ B_0 = \frac{\mu_0 I}{2R}\quad(1) $$
			Due to superposition, the magnetic induction at the center of one semicircle will be half of this value:
			$$ B_1 = B_2 = \frac{B_0}{2} = \frac{\mu_0 I}{4R}\quad(2) $$
			From the drawing, it is clear that the angle between the vectors $\vec{B}_1$ and $\vec{B}_2$ is equal to $\alpha$
			$$ (\vec{B_1},\vec{B_2})=\alpha $$
			The total vector of magnetic induction is equal to the vector sum of the components of magnetic induction of two rings $\vec{B}_1$ and $\vec{B}_2$
			$$ \vec{B} = \vec{B_1} + \vec{B_2} $$
			We will find the modulus of this vector using the cosine law.
			$$ B = \sqrt{{B_1}^2 + {B_2}^2 - 2B_1 B_2 \cos\alpha} $$
			Taking into account the expression $(2)$, we find the total magnetic induction as
			$$ B= \frac{\mu_0 I}{4R}\sqrt{2-2\cos\alpha} = \frac{\mu_0 I}{2R}\cdot\sin\frac{\alpha}{2}\quad(3) $$
			Dividing the expression $(3)$ by $(1)$, we obtain the desired ratio
			$$ \boxed{\frac{B}{B_0} = \sin\frac{\alpha}{2}} $$

			</p>

			<h4>Answer</h4>
			<p>
			$$\frac{B}{B_0} =\sin\frac{\alpha}{2}$$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Andrei Yersh<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> alex@savchenkosolutions.com <br></small>
			</p>
			</footer>
			</body>

			</html>