Long straight wires with current intersect at right angles. Determine the magnetic field induction at a point with coordinates x and y if the coordinate axes are the wires and the current in the wires is I.

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#9.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $9.2.6.$ Long straight wires with current intersect at right angles. Determine the magnetic field induction at a point with coordinates $x$ and $y$ if the coordinate axes are the wires and 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 $$\vec{B} = \vec{B_1} + \vec{B_2}$$ Using the right-hand rule, we can determine the direction of the magnetic induction lines and notice that $\vec{B_1}$ and $\vec{B_2}$ are directed perpendicular to the $xy$ plane, which means they are co-directed $$\boxed{B=B_1+B_2}\quad(1)$$ We will find the magnetic induction of each of them as the induction of an infinite single-horned wire. $$B_1 = \frac{\mu_0 I}{2\pi} \cdot \frac{1}{x};\quad B_2 = \frac{\mu_0 I}{2\pi} \cdot \frac{1}{y}\quad(2)$$ We substitute the obtained expression $(2)$ into $(1)$ $$\boxed{B = {B_1} +{B_2} = \frac{\mu_0 I}{2\pi}\cdot \left(\frac{1}{x}+\frac{1}{y}\right)}$$

    </p>

    <h4>Answer</h4>
    <p>
        $$B=\frac{\mu_0 I}{2\pi}\cdot \left(\frac{1}{x}+\frac{1}{y}\right)$$
    </p>
	<p style="text-align: right; font-style: italic; font-size: 14;">
      Andrei Yersh<br>
    </p>


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