$1.2.1.$ 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.
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
$$\fbox{$v_\text{av} = \frac{2}{\pi} \cdot v \cdot \frac{R - r}{R + r}$}$$
$ v_{\text{av}} = \frac{2}{\pi} v \frac{R - r}{R + r} $, directed along the interface