Translated 9.2.1-9.2.9
en/9.2.7.md
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| + | <title>Long straight wires with current intersect at an angle \alpha. Find the magnetic field induction on a straight line passing through the point of intersection of the wires perpendicular to both of them. The current in the wires is I.</title> | ||
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| + | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | ||
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| + | <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> | ||
| + | $9.2.7.$ Long straight wires with current intersect at an angle $\alpha$. Find the magnetic field induction on a straight line passing through the point of intersection of the wires perpendicular to both of them. The current in the wires is $I$. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/9/9.2.7/9.2.7_1.png" | ||
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| + | <figcaption> | ||
| + | Direction of magnetic induction created by current in wires | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | We find the total vector of magnetic induction as the sum of the vectors of magnetic induction created by each of the vectors | ||
| + | $$ \vec{B} = \vec{B_1} + \vec{B_2}\quad(1) $$ | ||
| + | According to the right-hand rule, we can determine the direction of the magnetic induction lines and notice that $\vec{B_1}$ and $\vec{B_2}$ lie in the same plane, parallel to the plane of the wires, at an angle of $\pi - \alpha$. Then, the total magnetic induction vector $\vec{B}$ from the expression $(1)$, then when adding through the cosine theorem will be an adjacent angle — $\alpha$ | ||
| + | $$ \boxed{B=\sqrt{B_1^2+B_2^2 - 2B_1B_2\cos\alpha}}\quad(2) $$ | ||
| + | We will find the magnetic induction of each of them as the induction of an infinite single-horned wire. | ||
| + | $$ B_1 = B_2 = \frac{\mu_0 I}{2\pi R}\quad(3) $$ | ||
| + | We substitute the obtained expression $(3)$ into $(2)$ and obtain the desired magnetic field induction | ||
| + | $$ B = \frac{\mu_0 I}{2\pi R} \sqrt{2-2\cos\alpha} \Rightarrow \boxed{B = \frac{\mu_0 I}{\pi R}\sin\left(\frac{\alpha}{2}\right)} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$B = \frac{\mu_0 I}{\pi R}\sin\left(\frac{\alpha}{2}\right)$$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
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| <meta name="description" content="Long straight wires with current intersect at an angle \alpha. Find the magnetic field induction on a straight line passing through the point of intersection of the wires perpendicular to both of them. The current in the wires is I."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="Long straight wires with current intersect at an angle \alpha. Find the magnetic field induction on a straight line passing through the point of intersection of the wires perpendicular to both of them. The current in the wires is I."> | |||
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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> | |||
| $9.2.7.$ Long straight wires with current intersect at an angle $\alpha$. Find the magnetic field induction on a straight line passing through the point of intersection of the wires perpendicular to both of them. The current in the wires is $I$. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/9/9.2.7/9.2.7_1.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| Direction of magnetic induction created by current in wires | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| We find the total vector of magnetic induction as the sum of the vectors of magnetic induction created by each of the vectors | |||
| $$ \vec{B} = \vec{B_1} + \vec{B_2}\quad(1) $$ | |||
| According to the right-hand rule, we can determine the direction of the magnetic induction lines and notice that $\vec{B_1}$ and $\vec{B_2}$ lie in the same plane, parallel to the plane of the wires, at an angle of $\pi - \alpha$. Then, the total magnetic induction vector $\vec{B}$ from the expression $(1)$, then when adding through the cosine theorem will be an adjacent angle — $\alpha$ | |||
| $$ \boxed{B=\sqrt{B_1^2+B_2^2 - 2B_1B_2\cos\alpha}}\quad(2) $$ | |||
| We will find the magnetic induction of each of them as the induction of an infinite single-horned wire. | |||
| $$ B_1 = B_2 = \frac{\mu_0 I}{2\pi R}\quad(3) $$ | |||
| We substitute the obtained expression $(3)$ into $(2)$ and obtain the desired magnetic field induction | |||
| $$ B = \frac{\mu_0 I}{2\pi R} \sqrt{2-2\cos\alpha} \Rightarrow \boxed{B = \frac{\mu_0 I}{\pi R}\sin\left(\frac{\alpha}{2}\right)} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$B = \frac{\mu_0 I}{\pi R}\sin\left(\frac{\alpha}{2}\right)$$ | |||
| </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> | |||