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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="../../#11.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#11.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $11.5.11^*.$ Calculate the inductance of a long solenoid of radius $r$ and length $l$ placed inside a long superconducting tube of radius $R$ along its axis. Number of turns of the solenoid $N$. | | $11.5.11^*.$ Calculate the inductance of a long solenoid of radius $r$ and length $l$ placed inside a long superconducting tube of radius $R$ along its axis. Number of turns of the solenoid $N$. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $11.5.11^*$ | | For problem $11.5.11^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw1.png" | | <img src="draw1.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| Direction of currents inside the solenoid | | Direction of currents inside the solenoid |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| The following expressions follow from the figure | | The following expressions follow from the figure |
| $$i_\perp=i\cos\alpha$$ | | $$i_\perp=i\cos\alpha$$ |
| $$i_\parallel=i\sin\alpha$$ | | $$i_\parallel=i\sin\alpha$$ |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw2.png" | | <img src="draw2.png" |
| loading="lazy" width="280" /> | | loading="lazy" width="280" /> |
| <figcaption> | | <figcaption> |
| Geometry of the solenoid | | Geometry of the solenoid |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| From the geometry of the solenoid, follows the aspect ratio | | From the geometry of the solenoid, follows the aspect ratio |
| $$\sin\alpha = \frac{r}{R}$$ | | $$\sin\alpha = \frac{r}{R}$$ |
| $$\cos\alpha = \sqrt{1-\frac{r^2}{R^2}}$$ | | $$\cos\alpha = \sqrt{1-\frac{r^2}{R^2}}$$ |
| Ampère's circuital law | | Ampère's circuital law |
| $$\oint B\,dl=\mu_0\sum_{k=1}^{N}i_k\Rightarrow B\cdot l=\mu_0 i_\perp N$$ | | $$\oint B\,dl=\mu_0\sum_{k=1}^{N}i_k\Rightarrow B\cdot l=\mu_0 i_\perp N$$ |
| By definition of the magnetic flux | | By definition of the magnetic flux |
| $$\left\{\begin{matrix} | | $$\left\{\begin{matrix} |
| \Phi=L i \\ | | \Phi=L i \\ |
| \Phi=NBS\cos\alpha | | \Phi=NBS\cos\alpha |
| \end{matrix}\right.$$ | | \end{matrix}\right.$$ |
| Equating the two equations | | Equating the two equations |
| $$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$ | | $$L i=\mu_0 i \cos\alpha \frac{N}{l}NS\cos\alpha$$ |
| $$L=\mu_0 \frac{N^2}{l}S\cos^2\alpha$$ | | $$L=\mu_0 \frac{N^2}{l}S\cos^2\alpha$$ |
| Given the formula for the area of a circle $S=\pi r^2$, we substitute the value of $\cos\alpha$ | | Given the formula for the area of a circle $S=\pi r^2$, we substitute the value of $\cos\alpha$ |
| $$\boxed{L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}}$$ | | $$\boxed{L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}$$ | | $$L=\mu_{0}\pi r^{2}\left(1-\frac{r^{2}}{R^{2}}\right)\frac{N^{2}}{l}$$ |
