The figure shows the trajectory of an electron that drifts along an interface plane of regions with different magnetic fields. Its trajectory consists of alternating semicircles of radius R and r. The velocity of the electron is constant modulo and equals v. Find the average velocity of the electron over a long period of time.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#1.2">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.2.1.$ The figure shows the trajectory of an electron that drifts along an interface plane of regions with different magnetic fields. Its trajectory consists of alternating semicircles of radius $R$ and $r$. The velocity of the electron is constant modulo and equals $v$. Find the average velocity of the electron over a long period of time.

    <h3>Solution</h3>
    <p>
        Consider the motion of an electron from point $ A $ to point $ B $. At these points the electron is in equal “phase”. The subsequent motion of the electron will be repeated.

Movement of an electron along the interface

$S = N \cdot (2R - 2r)$.

The time taken is equal to the sum of

$$t = N \cdot \left( \frac{\pi R}{v} + \frac{\pi r}{v} \right) = N \cdot \frac{\pi}{v} \cdot (R + r)$$

Average velocity of an electron over a large time interval

$$v_\text{av} = \frac{S}{t} = \frac{N \cdot 2(R - r)}{N \cdot \frac{\pi}{v}(R + r)}$$

after conversion

$$\fbox{$v_\text{av} = \frac{2}{\pi} \cdot v \cdot \frac{R - r}{R + r}$}$$

    <h4>Answer</h4>
    <p>
        $ v_{\text{av}} = \frac{2}{\pi} v \frac{R - r}{R + r} $, directed along the interface
    </p>


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