Translated 9.2.10-9.2.17

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

правка #11456 GitHub ef227c0

en/9.2.13.md +100 −0

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		+	<meta name="description" content="A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.">
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		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
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		+	<meta property="og:description" content="A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.">
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		+	<title>A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.</title>
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		+	<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>
		+	$9.2.13.$ A long straight wire with current $I$ has a section in the form of a semicircle of radius $R$. Determine the magnetic field induction at the center of the semicircle.
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/9/9.2.13/statement.png"
		+	loading="lazy" width="230" />
		+	<figcaption>
		+	For problem $9.2.13$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	Let's write down the expression for the induction of a section of wire
		+	$$ d\vec B = \frac{\mu_0 [\vec{(Idx)}\times \vec{R}]}{4\pi R^3} $$
		+	We find the modulus of the vector $d\vec B$ by transforming the vector product
		+	$$ dB = \frac{\mu_0 Idx}{4\pi R^2}\sin\alpha $$
		+	Where $\alpha$ is the angle between the radius vector and the direction of the current element. Given that they coincide, then $\alpha = 0$:
		+	$$ \sin\alpha = 0\Rightarrow dB = 0 $$
		+	For a straight section of wire, then such a design creates a field similar to the field of the semicircle, which we already calculated in <a href="../9.2.11">9.2.11</a>
		+	$$ \boxed{B = \frac{\mu_0 I}{4R}} $$
		+
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$B=\frac{\mu_0 I}{4R}$$
		+	</p>
		+	<p style="text-align: right; font-style: italic; font-size: 14;">
		+	Andrei Yersh<br>
		+	</p>
		+
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			<meta http-equiv="content-language" content="en">
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			<meta name="description" content="A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>A long straight wire with current I has a section in the form of a semicircle of radius R. Determine the magnetic field induction at the center of the semicircle.</title>
			<link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css">
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			<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>
			$9.2.13.$ A long straight wire with current $I$ has a section in the form of a semicircle of radius $R$. Determine the magnetic field induction at the center of the semicircle.
			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/9/9.2.13/statement.png"
			loading="lazy" width="230" />
			<figcaption>
			For problem $9.2.13$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>
			Let's write down the expression for the induction of a section of wire
			$$ d\vec B = \frac{\mu_0 [\vec{(Idx)}\times \vec{R}]}{4\pi R^3} $$
			We find the modulus of the vector $d\vec B$ by transforming the vector product
			$$ dB = \frac{\mu_0 Idx}{4\pi R^2}\sin\alpha $$
			Where $\alpha$ is the angle between the radius vector and the direction of the current element. Given that they coincide, then $\alpha = 0$:
			$$ \sin\alpha = 0\Rightarrow dB = 0 $$
			For a straight section of wire, then such a design creates a field similar to the field of the semicircle, which we already calculated in <a href="../9.2.11">9.2.11</a>
			$$ \boxed{B = \frac{\mu_0 I}{4R}} $$

			</p>

			<h4>Answer</h4>
			<p>
			$$B=\frac{\mu_0 I}{4R}$$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Andrei Yersh<br>
			</p>


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