<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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.">
<meta property="og:title" 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.">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og: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.">
<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>
$1.2.1.$ 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>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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.">
<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.">
<meta property="og:title" 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.">
<meta property="og:title" 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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og: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.">
<meta property="og: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.">
<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>
<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>
$1.2.1.$ 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.
$1.2.1.$ 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>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
For problem $1.2.1$
For problem $1.2.1$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<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.
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>
<p>
<p>
Movement of an electron along the interface
Movement of an electron along the interface
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$ S = N \cdot (2R - 2r) $.
$ S = N \cdot (2R - 2r) $.
</p>
</p>
<p>
<p>
The time taken is equal to the sum of
The time taken is equal to the sum of
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$ t = N \cdot\left( \frac{\pi R}{v} + \frac{\pi r}{v}\right) = N \cdot\frac{\pi}{v}\cdot (R + r) $$
$$ t = N \cdot\left( \frac{\pi R}{v} + \frac{\pi r}{v}\right) = N \cdot\frac{\pi}{v}\cdot (R + r) $$
</p>
</p>
<p>
<p>
Average velocity of an electron over a large time interval</p>
Average velocity of an electron over a large time interval</p>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
