Translated 9.2.1-9.2.9
en/9.2.9.md
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| + | <title>Prove that at large distances from two series-connected sections of wire l_1 and l_2, through which current flows, the magnetic field is close to the magnetic field of the section of wire \vec{l} = \vec{l_1} + \vec{l_2} through which the same current flows.</title> | ||
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| + | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | ||
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| + | Solutions of Savchenko Problems in Physics <br> | ||
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| + | <h3 id="back-link"><a href="../../#9.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $9.2.9.$ Prove that at large distances from two series-connected sections of wire $l_1$ and $l_2$, through which current flows, the magnetic field is close to the magnetic field of the section of wire $\vec{l} = \vec{l_1} + \vec{l_2}$ through which the same current flows. | ||
| + | </p> | ||
| + | |||
| + | <h3>Proof</h3> | ||
| + | <p> | ||
| + | Let's find the magnetic field from each of the wires in vector form: | ||
| + | $$ \boxed{\vec{B} = \vec{B_1} + \vec{B_2}}\quad(1) $$ | ||
| + | Where are the components of each magnetic induction | ||
| + | $$ \vec{B_1} = \frac{\mu_0}{4\pi R^3}[\vec{Il_1} \times \vec{R}] = \frac{\mu_0 I}{4\pi R^3}[\vec{l_1}\times \vec{R}]\quad(2) $$ | ||
| + | |||
| + | $$ \vec{B_2} = \frac{\mu_0}{4\pi R^3}[\vec{Il_2} \times \vec{R}] = \frac{\mu_0 I}{4\pi R^3}[\vec{l_2}\times \vec{R}]\quad(3) $$ | ||
| + | We substitute the values $(2)$ and $(3)$ into $(1)$, | ||
| + | $$ \vec{B} = \frac{\mu_0 I}{4\pi R^3}[(\vec{l_1} + \vec{l_2})\times \vec{R}] \quad(4) $$ | ||
| + | The last transformation follows from the property of distributivity under addition for the vector product. </p><p> | ||
| + | Thus, we have obtained that in sum the two wires have the same effect as the wire | ||
| + | $$ \vec{l} = \vec{l_1} + \vec{l_2} $$ | ||
| + | </p> | ||
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| <meta name="description" content="Prove that at large distances from two series-connected sections of wire l_1 and l_2, through which current flows, the magnetic field is close to the magnetic field of the section of wire \vec{l} = \vec{l_1} + \vec{l_2} through which the same current flows."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="Prove that at large distances from two series-connected sections of wire l_1 and l_2, through which current flows, the magnetic field is close to the magnetic field of the section of wire \vec{l} = \vec{l_1} + \vec{l_2} through which the same current flows."> | |||
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| <meta property="og:description" content="Prove that at large distances from two series-connected sections of wire l_1 and l_2, through which current flows, the magnetic field is close to the magnetic field of the section of wire \vec{l} = \vec{l_1} + \vec{l_2} through which the same current flows."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>Prove that at large distances from two series-connected sections of wire l_1 and l_2, through which current flows, the magnetic field is close to the magnetic field of the section of wire \vec{l} = \vec{l_1} + \vec{l_2} through which the same current flows.</title> | |||
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| <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> | |||
| $9.2.9.$ Prove that at large distances from two series-connected sections of wire $l_1$ and $l_2$, through which current flows, the magnetic field is close to the magnetic field of the section of wire $\vec{l} = \vec{l_1} + \vec{l_2}$ through which the same current flows. | |||
| </p> | |||
| <h3>Proof</h3> | |||
| <p> | |||
| Let's find the magnetic field from each of the wires in vector form: | |||
| $$ \boxed{\vec{B} = \vec{B_1} + \vec{B_2}}\quad(1) $$ | |||
| Where are the components of each magnetic induction | |||
| $$ \vec{B_1} = \frac{\mu_0}{4\pi R^3}[\vec{Il_1} \times \vec{R}] = \frac{\mu_0 I}{4\pi R^3}[\vec{l_1}\times \vec{R}]\quad(2) $$ | |||
| $$ \vec{B_2} = \frac{\mu_0}{4\pi R^3}[\vec{Il_2} \times \vec{R}] = \frac{\mu_0 I}{4\pi R^3}[\vec{l_2}\times \vec{R}]\quad(3) $$ | |||
| We substitute the values $(2)$ and $(3)$ into $(1)$, | |||
| $$ \vec{B} = \frac{\mu_0 I}{4\pi R^3}[(\vec{l_1} + \vec{l_2})\times \vec{R}] \quad(4) $$ | |||
| The last transformation follows from the property of distributivity under addition for the vector product. </p><p> | |||
| Thus, we have obtained that in sum the two wires have the same effect as the wire | |||
| $$ \vec{l} = \vec{l_1} + \vec{l_2} $$ | |||
| </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> | |||
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