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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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>
        $9.2.13.$ 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.

    <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 9.2.11
$$\boxed{B = \frac{\mu_0 I}{4R}}$$

    </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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