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| + | <meta name="description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?"> | ||
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| + | <title>Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?</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="../../#2.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.3.38.$ Potential energy of electrostatic interaction of point charges $q$ and $Q$ located at a distance $r$ from each other, $U = \frac{kqQ}{r}$. Find the electrostatic force. Which charges have repulsion and which have attraction? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Let us find the derivative of the potential energy of interaction by distance | ||
| + | $$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$ | ||
| + | $$dU=-\frac{kqQ}{r^2}dr\quad(1)$$ | ||
| + | Also by the law of conservation of energy, the change in that energy is equal to the work to be done | ||
| + | $$dU=dA\Rightarrow dU=-Fdr\quad(2)$$ | ||
| + | <i>NO: The larger $r$ is, the smaller $F$ is</i> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="draw.png" | ||
| + | loading="lazy" width="270" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the system | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | According to Newton's third law | ||
| + | $$\vec{F}_1=-\vec{F}_2=F$$ | ||
| + | Substituting $(2)$ into $(1)$ leads to | ||
| + | $$-\frac{kqQ}{r^2}dr=-Fdr$$ | ||
| + | $$\boxed{F=\frac{kqQ}{r^2}}$$ | ||
| + | When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction. | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$F=\frac{kqQ}{r^2}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
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| + | <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> astrosander01@gmail.com <br></small> | ||
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| @@ -0,0 +1,97 @@ | |||
| <!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="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>Potential energy of electrostatic interaction of point charges q and Q located at a distance r from each other, U = \frac{kqQ}{r}. Find the electrostatic force. Which charges have repulsion and which have attraction?</title> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <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="../../#2.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.3.38.$ Potential energy of electrostatic interaction of point charges $q$ and $Q$ located at a distance $r$ from each other, $U = \frac{kqQ}{r}$. Find the electrostatic force. Which charges have repulsion and which have attraction? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Let us find the derivative of the potential energy of interaction by distance | |||
| $$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$ | |||
| $$dU=-\frac{kqQ}{r^2}dr\quad(1)$$ | |||
| Also by the law of conservation of energy, the change in that energy is equal to the work to be done | |||
| $$dU=dA\Rightarrow dU=-Fdr\quad(2)$$ | |||
| <i>NO: The larger $r$ is, the smaller $F$ is</i> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="draw.png" | |||
| loading="lazy" width="270" /> | |||
| <figcaption> | |||
| Forces acting on the system | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| According to Newton's third law | |||
| $$\vec{F}_1=-\vec{F}_2=F$$ | |||
| Substituting $(2)$ into $(1)$ leads to | |||
| $$-\frac{kqQ}{r^2}dr=-Fdr$$ | |||
| $$\boxed{F=\frac{kqQ}{r^2}}$$ | |||
| When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction. | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$F=\frac{kqQ}{r^2}$$ | |||
| </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> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||