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| <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 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $2.3.38.$ 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> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let us find the derivative of the potential energy of interaction by distance | | Let us find the derivative of the potential energy of interaction by distance |
| $$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$ | | $$\frac{dU}{dr}=\left(\frac{kqQ}{r}\right)'$$ |
| $$dU=-\frac{kqQ}{r^2}dr\quad(1)$$ | | $$dU=-\frac{kqQ}{r^2}dr\quad(1)$$ |
| Also by the law of conservation of energy, the change in that energy is equal to the work to be done | | Also by the law of conservation of energy, the change in that energy is equal to the work to be done |
| $$dU=dA\Rightarrow dU=-Fdr\quad(2)$$ | | $$dU=dA\Rightarrow dU=-Fdr\quad(2)$$ |
| <i>NO: The larger $r$ is, the smaller $F$ is</i> | | <i>NO: The larger $r$ is, the smaller $F$ is</i> |
| </p> | | </p> |
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| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" width="270" /> | | loading="lazy" width="270" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system | | Forces acting on the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| According to Newton's third law | | According to Newton's third law |
| $$\vec{F}_1=-\vec{F}_2=F$$ | | $$\vec{F}_1=-\vec{F}_2=F$$ |
| Substituting $(2)$ into $(1)$ leads to | | Substituting $(2)$ into $(1)$ leads to |
| $$-\frac{kqQ}{r^2}dr=-Fdr$$ | | $$-\frac{kqQ}{r^2}dr=-Fdr$$ |
| $$\boxed{F=\frac{kqQ}{r^2}}$$ | | $$\boxed{F=\frac{kqQ}{r^2}}$$ |
| When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction. | | When $qQ > 0$ — repulsion, when $qQ < 0$ — attraction. |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$F=\frac{kqQ}{r^2}$$ | | $$F=\frac{kqQ}{r^2}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </p> | | </p> |
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