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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
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| <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."> |
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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> | | <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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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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. | | $1.2.1.$ 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. |
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| <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> |
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| <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> |
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| <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> |
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| <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> |
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| <p> | | <p> |
| Average velocity of an electron over a large time interval</p> | | Average velocity of an electron over a large time interval</p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$ v_\text{av} = \frac{S}{t} = \frac{N \cdot 2(R - r)}{N \cdot \frac{\pi}{v}(R + r)} $$ | | $$ v_\text{av} = \frac{S}{t} = \frac{N \cdot 2(R - r)}{N \cdot \frac{\pi}{v}(R + r)} $$ |
| </p> | | </p> |
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| <p> | | <p> |
| after conversion | | after conversion |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\fbox{$v_\text{av} = \frac{2}{\pi} \cdot v \cdot \frac{R - r}{R + r}$}$$ | | $$\fbox{$v_\text{av} = \frac{2}{\pi} \cdot v \cdot \frac{R - r}{R + r}$}$$ |
| </p> | | </p> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $ v_{\text{av}} = \frac{2}{\pi} v \frac{R - r}{R + r} $, directed along the interface | | $ v_{\text{av}} = \frac{2}{\pi} v \frac{R - r}{R + r} $, directed along the interface |
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