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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="../../#6.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#6.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $6.5.2$ | | $6.5.2$ |
| Two electroconductor pistons of cross section $S$ form a plane capacitor in a tube of an isolating material which contains air at atmospheric pressure $P_0$. How changes the distance between pistons if charges of different signs $\pm Q$ are applied to them? The system is good heat conductor and there is no friction. | | Two electroconductor pistons of cross section $S$ form a plane capacitor in a tube of an isolating material which contains air at atmospheric pressure $P_0$. How changes the distance between pistons if charges of different signs $\pm Q$ are applied to them? The system is good heat conductor and there is no friction. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" alt="6.5.2" width="200" /> | | loading="lazy" alt="6.5.2" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem 6.5.2 | | For problem 6.5.2 |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let's consider the following figure | | Let's consider the following figure |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" alt="6.5.2" width="450" /> | | loading="lazy" alt="6.5.2" width="450" /> |
| <figcaption> | | <figcaption> |
| Forces over pistons | | Forces over pistons |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Applying Newton Second Law over one of the piston, considering the pistons are massless, | | Applying Newton Second Law over one of the piston, considering the pistons are massless, |
| $$F_p = F_{p0} + F_e$$ | | $$F_p = F_{p0} + F_e$$ |
| where $F_p$, $F_{p0}$ and $F_e$ are forces due to enclosed air between pistons, due to atmospheric pressure and due to electric field, respectively. | | where $F_p$, $F_{p0}$ and $F_e$ are forces due to enclosed air between pistons, due to atmospheric pressure and due to electric field, respectively. |
| $$P = P_0 + \frac{F_e}{S} \;(1)$$ | | $$P = P_0 + \frac{F_e}{S} \;(1)$$ |
| The electric force $F_e$ is defined from the electric field generated by a charged plate that is $E = \frac{\sigma}{2\varepsilon_0}=\frac{Q}{2\varepsilon_0 S}$, so | | The electric force $F_e$ is defined from the electric field generated by a charged plate that is $E = \frac{\sigma}{2\varepsilon_0}=\frac{Q}{2\varepsilon_0 S}$, so |
| $$F_e = Q E =\frac{Q^2}{2\varepsilon_0 S} \;(2)$$ | | $$F_e = Q E =\frac{Q^2}{2\varepsilon_0 S} \;(2)$$ |
| Putting $(2)$ into $(1)$ | | Putting $(2)$ into $(1)$ |
| $$P = P_0 + \frac{Q^2}{2\varepsilon_0 S} \;(3)$$ | | $$P = P_0 + \frac{Q^2}{2\varepsilon_0 S} \;(3)$$ |
| Taking account that the system is good heat conductor, we can suppose that thermal equilibrium is instantaneously established or that the container acts as a thermos, keeping the temperature unchanged.Then, applying Boyle-Mariotte law | | Taking account that the system is good heat conductor, we can suppose that thermal equilibrium is instantaneously established or that the container acts as a thermos, keeping the temperature unchanged.Then, applying Boyle-Mariotte law |
| $$P_0 S L = P S l$$ | | $$P_0 S L = P S l$$ |
| $$\frac{L}{l} = \frac{P}{P_0} \;(4)$$ | | $$\frac{L}{l} = \frac{P}{P_0} \;(4)$$ |
| Substituting $(3)$ into $(4)$ | | Substituting $(3)$ into $(4)$ |
| $$\boxed{\frac{L}{l} = 1 + \frac{Q^2}{2P_0\varepsilon_0 S^2}}$$ | | $$\boxed{\frac{L}{l} = 1 + \frac{Q^2}{2P_0\varepsilon_0 S^2}}$$ |
| </p> | | </p> |
| | | |
| | | |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
| </p> | | </p> |
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