| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <meta http-equiv="content-language" content="en"> |
| <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="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> | | <meta name="description" content="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> | | <meta property="og:title" content="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> | | <meta property="og:description" content="Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0.</title> | | <title>Determine what the accelerating potential difference V should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates R_1 and R_2. Potential difference between the plates V_0.</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> |
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> |
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> |
| <script type="text/x-mathjax-config"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $7.1.12^*.$ Determine what the accelerating potential difference $V$ should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates $R_1$ and $R_2$. Potential difference between the plates $V_0$. | | $7.1.12^*.$ Determine what the accelerating potential difference $V$ should be in order for the electrons to follow the path shown in the figure. Radii of cylindrical capacitor plates $R_1$ and $R_2$. Potential difference between the plates $V_0$. |
| </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 $7.1.12^*$ | | For problem $7.1.12^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Gauss theorem | | Gauss theorem |
| $$\oint_{2\pi r}E\,dl=\frac{q}{\varepsilon_0}\Rightarrow E=\frac{q}{2\pi \varepsilon_0 r}$$ | | $$\oint_{2\pi r}E\,dl=\frac{q}{\varepsilon_0}\Rightarrow E=\frac{q}{2\pi \varepsilon_0 r}$$ |
| Let's write down the small change of the electrostatic field and then integrate it | | Let's write down the small change of the electrostatic field and then integrate it |
| $$dU=E \,dr\Rightarrow \int dU=\int E \,dr$$ | | $$dU=E \,dr\Rightarrow \int dU=\int E \,dr$$ |
| Integrate from $R_1$ to $R_2$ | | Integrate from $R_1$ to $R_2$ |
| $$U_0=\frac{q}{2\pi \varepsilon_0 r}\int_{R_1}^{R_2}\frac{dr}{r}=\frac{q}{2\pi \varepsilon_0 r}\ln\frac{R_2}{R_1}$$ | | $$U_0=\frac{q}{2\pi \varepsilon_0 r}\int_{R_1}^{R_2}\frac{dr}{r}=\frac{q}{2\pi \varepsilon_0 r}\ln\frac{R_2}{R_1}$$ |
| Law of conservation of energy | | Law of conservation of energy |
| $$\frac{mv^2}{2}=eU$$ | | $$\frac{mv^2}{2}=eU$$ |
| $$e\frac{q}{2\pi \varepsilon_0 \frac{R_1+R_2}{2}}=\frac{mv^2}{(\frac{R_1+R_2}{2})}$$ | | $$e\frac{q}{2\pi \varepsilon_0 \frac{R_1+R_2}{2}}=\frac{mv^2}{(\frac{R_1+R_2}{2})}$$ |
| From where we get | | From where we get |
| $$\frac{q}{2\pi \varepsilon_0}=\frac{mv^2}{e}$$ | | $$\frac{q}{2\pi \varepsilon_0}=\frac{mv^2}{e}$$ |
| Substituting the previously obtained values | | Substituting the previously obtained values |
| $$\frac{U_0}{\ln\frac{R_2}{R_1}}=\frac{2eU}{e}$$ | | $$\frac{U_0}{\ln\frac{R_2}{R_1}}=\frac{2eU}{e}$$ |
| From where | | From where |
| $$U=\frac{U_0}{2\ln\frac{R_2}{R_1}}$$ | | $$U=\frac{U_0}{2\ln\frac{R_2}{R_1}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$V=(V_0/2)/\ln(R_2/R_1)$$ | | $$V=(V_0/2)/\ln(R_2/R_1)$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Yuldashev Ulugbek<br> | | Yuldashev Ulugbek<br> |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <p> | | <p> |
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> |
| </p> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
