Added Luis's English solution of 6.2.11
en/6.2.11.md
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| + | <!DOCTYPE html> | ||
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| + | <meta name="description" content=""> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | </head> | ||
| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <div id = "logo"> | ||
| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $6.2.11$ Two infinite plates of thickness h are charged uniformly in volume and stacked together. The bulk charge density of the first plate is $\rho$, and the second plate -$\rho$. Find the maximum electric field intensity. | ||
| + | </p> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="statement.png" | ||
| + | loading="lazy" alt="6.2.11" width="350" /> | ||
| + | <figcaption> | ||
| + | For problem 6.2.11 | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Absolute value of total charge of each plate is $\rho S h$, where $S$ is the surface of plate's biggest faces. So, applying Gauss Law, the absolute value of electric field intensity generated by this plate is: | ||
| + | </p> | ||
| + | <p> | ||
| + | $$E = \frac{\rho h}{\varepsilon_0}$$ | ||
| + | </p> | ||
| + | <p> | ||
| + | Then, field generated by one of these biggest faces is the half of $E$. Let's consider now a point inside of one of these infinite plates. The resultant field over this point is the vectorial sum of the fields generated by lower negative plate, a plate with thickness $x$ (where $x$ is the distance from boundary between plates and the point) and a upper plate with thickness $h-x$: | ||
| + | </p> | ||
| + | <p> | ||
| + | $$E_R(x) = \frac{\rho h}{2\varepsilon_0} + \frac{\rho (h-x)}{2\varepsilon_0} - \frac{\rho x}{2\varepsilon_0}$$ | ||
| + | </p> | ||
| + | <p> | ||
| + | $$E_R(x) = \frac{\rho}{\varepsilon_0}(h-x)$$ | ||
| + | </p> | ||
| + | <p> | ||
| + | Hence, maximum field intensity is achieved when $x=0$, i.e., point is in boundary between plates. | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$E_{max} = \frac{\rho h}{\varepsilon_0}$$ | ||
| + | </p> | ||
| + | |||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | BSc. Luis Daniel Fernández Quintana<br> | ||
| + | Physics Department (FCNE)<br> | ||
| + | Universidad de Oriente, Cuba<br> | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <footer class="row container"> | ||
| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <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> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,107 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <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="description" content=""> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content=""> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content=""> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title></title> | |||
| <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"> | |||
| <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 type="text/x-mathjax-config"> | |||
| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
| jax: ['input/TeX', 'output/HTML-CSS'], | |||
| tex2jax: { | |||
| inlineMath: [['$', '$'], ['$', '$']], | |||
| processEscapes: true, | |||
| processClass: 'tex2jax', | |||
| ignoreClass: 'html' | |||
| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <div id = "logo"> | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $6.2.11$ Two infinite plates of thickness h are charged uniformly in volume and stacked together. The bulk charge density of the first plate is $\rho$, and the second plate -$\rho$. Find the maximum electric field intensity. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="statement.png" | |||
| loading="lazy" alt="6.2.11" width="350" /> | |||
| <figcaption> | |||
| For problem 6.2.11 | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Absolute value of total charge of each plate is $\rho S h$, where $S$ is the surface of plate's biggest faces. So, applying Gauss Law, the absolute value of electric field intensity generated by this plate is: | |||
| </p> | |||
| <p> | |||
| $$E = \frac{\rho h}{\varepsilon_0}$$ | |||
| </p> | |||
| <p> | |||
| Then, field generated by one of these biggest faces is the half of $E$. Let's consider now a point inside of one of these infinite plates. The resultant field over this point is the vectorial sum of the fields generated by lower negative plate, a plate with thickness $x$ (where $x$ is the distance from boundary between plates and the point) and a upper plate with thickness $h-x$: | |||
| </p> | |||
| <p> | |||
| $$E_R(x) = \frac{\rho h}{2\varepsilon_0} + \frac{\rho (h-x)}{2\varepsilon_0} - \frac{\rho x}{2\varepsilon_0}$$ | |||
| </p> | |||
| <p> | |||
| $$E_R(x) = \frac{\rho}{\varepsilon_0}(h-x)$$ | |||
| </p> | |||
| <p> | |||
| Hence, maximum field intensity is achieved when $x=0$, i.e., point is in boundary between plates. | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$E_{max} = \frac{\rho h}{\varepsilon_0}$$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| BSc. Luis Daniel Fernández Quintana<br> | |||
| Physics Department (FCNE)<br> | |||
| Universidad de Oriente, Cuba<br> | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <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> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||