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.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <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>

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>