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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 |
| $8.3.15^*$ | | $8.3.15^*$ |
| The attenuator is a voltage divider, the circuit of which is shown in the figure. What should be the resistances $R_1$ and $R_2$ so that at each subsequent resistance $R_1$ the voltage is ten times less than at the previous one? | | The attenuator is a voltage divider, the circuit of which is shown in the figure. What should be the resistances $R_1$ and $R_2$ so that at each subsequent resistance $R_1$ the voltage is ten times less than at the previous one? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" alt="8.3.15" width="300" /> | | loading="lazy" alt="8.3.15" width="300" /> |
| <figcaption> | | <figcaption> |
| For problem 8.3.15 | | For problem 8.3.15 |
| </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="8.3.15" width="450" /> | | loading="lazy" alt="8.3.15" width="450" /> |
| <figcaption> | | <figcaption> |
| Circuit analysis | | Circuit analysis |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Applying Kirchhoff Second Law (Loop's law). For loop $I$ | | Applying Kirchhoff Second Law (Loop's law). For loop $I$ |
| $$-i_{11}R_1-i_{22}R_2+i_{21}R_2=0$$ | | $$-i_{11}R_1-i_{22}R_2+i_{21}R_2=0$$ |
| $$-i_{11}R_1+R_2(i_{21}-i_{22})=0 \;(1)$$ | | $$-i_{11}R_1+R_2(i_{21}-i_{22})=0 \;(1)$$ |
| As $i_{11}R_1 = 10 i_{12}R_1$, that is, $i_{1n} = 10 i_{1(n+1)}$, hence | | As $i_{11}R_1 = 10 i_{12}R_1$, that is, $i_{1n} = 10 i_{1(n+1)}$, hence |
| $$i_{1n} = \frac{i_{11}}{10^{n-1}} \;\forall~n\geq2 \;(2)$$ | | $$i_{1n} = \frac{i_{11}}{10^{n-1}} \;\forall~n\geq2 \;(2)$$ |
| For loop $II$ and according $(2)$ | | For loop $II$ and according $(2)$ |
| $$-i_{N+1}(R_1+r)+i_{2N}R_2=0$$ | | $$-i_{N+1}(R_1+r)+i_{2N}R_2=0$$ |
| $$-\frac{i_{11}}{10^{N-1}}(R_1+r)+i_{2N}R_2=0 \;(3)$$ | | $$-\frac{i_{11}}{10^{N-1}}(R_1+r)+i_{2N}R_2=0 \;(3)$$ |
| For subsequent bifurcations (applying Kirchhoff First Law) | | For subsequent bifurcations (applying Kirchhoff First Law) |
| $$i_{11} = i_{12}+i_{22}$$ | | $$i_{11} = i_{12}+i_{22}$$ |
| $$i_{22} = i_{11}-i_{12}$$ | | $$i_{22} = i_{11}-i_{12}$$ |
| So, | | So, |
| $$i_{2n} = i_{1(n-1)}-i_{1n} \;\forall~n\geq2 \;(4)$$ | | $$i_{2n} = i_{1(n-1)}-i_{1n} \;\forall~n\geq2 \;(4)$$ |
| According $(2)$, let's modify $(4)$ | | According $(2)$, let's modify $(4)$ |
| $$i_{2n} = \frac{i_{11}}{10^{n-2}}-\frac{i_{11}}{10^{n-1}}$$ | | $$i_{2n} = \frac{i_{11}}{10^{n-2}}-\frac{i_{11}}{10^{n-1}}$$ |
| $$i_{2n} = \frac{9i_{11}}{10^{n-1}} \;(5)$$ | | $$i_{2n} = \frac{9i_{11}}{10^{n-1}} \;(5)$$ |
| According $(5)$ into $(3)$ | | According $(5)$ into $(3)$ |
| $$9R_2 = R_1+r \;(6)$$ | | $$9R_2 = R_1+r \;(6)$$ |
| According $(5)$, from $(1)$ | | According $(5)$, from $(1)$ |
| $$R_1 = \frac{81}{10} R_2 \;(7)$$ | | $$R_1 = \frac{81}{10} R_2 \;(7)$$ |
| Finally, solving the equation system formed by $(6)$ and $(7)$, it is obtained | | Finally, solving the equation system formed by $(6)$ and $(7)$, it is obtained |
| $$\boxed{R_1 = 9r}$$ | | $$\boxed{R_1 = 9r}$$ |
| and, | | and, |
| $$\boxed{R_2 = \frac{10}{9}r}$$ | | $$\boxed{R_2 = \frac{10}{9}r}$$ |
| </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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