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en/8.4.5.md
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| + | ### Statement | ||
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| + | $8.4.5.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
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| + |  | ||
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| + | Initially, there are charge $q=CV$ and energy $CV^2/2$ stored on the left capacitor. When the circuit is closed, the same current flows through both resistors and the ratio of the heat generated in each of them is | ||
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| + | \[\frac{Q_1}{Q_2}=\frac{R_1}{R_2}.\] | ||
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| + | When the circuit reaches steady state and there is no current, there are charge $q/2$ and energy $(q/2)^2/(2C)=CV^2/8$ stored on each capacitor. Thus, the total heat generated is | ||
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| + | \[Q_1+Q_2=\frac{1}{2}CV^2-2\cdot\frac{1}{8}CV^2=\frac{1}{4}CV^2.\] | ||
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| + | Solving both displayed equations for $Q_1$ and $Q_2$, we have | ||
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| + | \[Q_1=\frac{1}{4}CV^2\cdot\frac{R_1}{R_1+R_2}\qquad\mbox\qquad Q_2=\frac{1}{4}CV^2\cdot\frac{R_2}{R_1+R_2}.\] | ||
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| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $8.4.5.$ [Insert the problem statement] | |||
| ### Solution | |||
|  | |||
| Initially, there are charge $q=CV$ and energy $CV^2/2$ stored on the left capacitor. When the circuit is closed, the same current flows through both resistors and the ratio of the heat generated in each of them is | |||
| \[\frac{Q_1}{Q_2}=\frac{R_1}{R_2}.\] | |||
| When the circuit reaches steady state and there is no current, there are charge $q/2$ and energy $(q/2)^2/(2C)=CV^2/8$ stored on each capacitor. Thus, the total heat generated is | |||
| \[Q_1+Q_2=\frac{1}{2}CV^2-2\cdot\frac{1}{8}CV^2=\frac{1}{4}CV^2.\] | |||
| Solving both displayed equations for $Q_1$ and $Q_2$, we have | |||
| \[Q_1=\frac{1}{4}CV^2\cdot\frac{R_1}{R_1+R_2}\qquad\mbox\qquad Q_2=\frac{1}{4}CV^2\cdot\frac{R_2}{R_1+R_2}.\] | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||