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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    <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.

Thus, we have obtained that in sum the two wires have the same effect as the wire
$$\vec{l} = \vec{l_1} + \vec{l_2}$$

Andrei Yersh

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