Правка разделов «Statement», «Answer»
en/11.3.12.md
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| @@ -1,6 +1,8 @@ | |||
| ### Statement | |||
| − | $11.3.12.$ | ||
| + | $11.3.12.$ | ||
| + | All dimensions of the conductor were increased by k times. How many times | ||
| + | will the inductance of the conductor change? | ||
| ### Solution | |||
| For this problem we can use scaling arguments. The inductance of a system of conductors with linear media is proportional to the characteristic length. | |||
| If each linear dimension is multiplied by k, the total inductance is also multiplied by k. This follows from dimensional analysis: | |||
| inductance has units of | |||
| $\text{H} = \text{Wb/A} = \text{T·m}^2/\text{A}$ | |||
| and since the magnetic field of a current configuration scales inversely with distance, the magnetic flux ends up being proportional to the current and to the length, so that L is proportional to length. | |||
| Therefore, when all linear dimensions (radii, distances between conductors, and total length of the system) are scaled by a factor k, the inductance increases by a factor of k: | |||
| $\boxed{L_{\text{new}} = k\,L_{\text{original}}}$. | |||
| @@ -19,4 +21,4 @@Solution | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $\boxed{L_{\text{new}} = k\,L_{\text{original}}}$. | ||
| @@ -1,6 +1,8 @@ | |||
| ### Statement | ### Statement | ||
| $11.3.12.$ |
$11.3.12.$ | ||
| All dimensions of the conductor were increased by k times. How many times | |||
| will the inductance of the conductor change? | |||
| ### Solution | ### Solution | ||
| For this problem we can use scaling arguments. The inductance of a system of conductors with linear media is proportional to the characteristic length. | For this problem we can use scaling arguments. The inductance of a system of conductors with linear media is proportional to the characteristic length. | ||
| If each linear dimension is multiplied by k, the total inductance is also multiplied by k. This follows from dimensional analysis: | If each linear dimension is multiplied by k, the total inductance is also multiplied by k. This follows from dimensional analysis: | ||
| inductance has units of | inductance has units of | ||
| $\text{H} = \text{Wb/A} = \text{T·m}^2/\text{A}$ | $\text{H} = \text{Wb/A} = \text{T·m}^2/\text{A}$ | ||
| and since the magnetic field of a current configuration scales inversely with distance, the magnetic flux ends up being proportional to the current and to the length, so that L is proportional to length. | and since the magnetic field of a current configuration scales inversely with distance, the magnetic flux ends up being proportional to the current and to the length, so that L is proportional to length. | ||
| Therefore, when all linear dimensions (radii, distances between conductors, and total length of the system) are scaled by a factor k, the inductance increases by a factor of k: | Therefore, when all linear dimensions (radii, distances between conductors, and total length of the system) are scaled by a factor k, the inductance increases by a factor of k: | ||
| $\boxed{L_{\text{new}} = k\,L_{\text{original}}}$. | $\boxed{L_{\text{new}} = k\,L_{\text{original}}}$. | ||
| @@ -19,4 +21,4 @@Solution | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $\boxed{L_{\text{new}} = k\,L_{\text{original}}}$. | ||