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+ <meta name="description" content="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.">
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+ <title>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.</title>
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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Problems in Physics</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <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.
+
+</p>
+<center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" width="230" />
+ <figcaption>
+ For problem $1.2.1$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <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.
+</p>
+
+<p>
+ Movement of an electron along the interface
+</p>
+<p style="text-align: center;">
+ $ S = N \cdot (2R - 2r) $.
+</p>
+
+<p>
+ The time taken is equal to the sum of
+</p>
+<p style="text-align: center;">
+ $$ t = N \cdot \left( \frac{\pi R}{v} + \frac{\pi r}{v} \right) = N \cdot \frac{\pi}{v} \cdot (R + r) $$
+</p>
+
+<p>
+ Average velocity of an electron over a large time interval</p>
+<p style="text-align: center;">
+ $$ v_\text{av} = \frac{S}{t} = \frac{N \cdot 2(R - r)}{N \cdot \frac{\pi}{v}(R + r)} $$
+</p>
+
+<p>
+ after conversion
+</p>
+<p style="text-align: center;">
+ $$\fbox{$v_\text{av} = \frac{2}{\pi} \cdot v \cdot \frac{R - r}{R + r}$}$$
+</p>
+ </p>
+
+ <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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