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en/11.2.8.md
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| + | ### Statement | ||
| + | |||
| + | $11.2.8.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
| + | |||
| + | a) Well, everything is pretty trivial here, the voltage on each capacitor: | ||
| + | |||
| + | $$U = \frac{\varphi}{2}$$ | ||
| + | |||
| + | So there is a charge on each capacitor: | ||
| + | |||
| + | $$q_1 = C_1 \frac{\varphi}{2}$$ | ||
| + | |||
| + | $$q_2 = C_2 \frac{\varphi}{2}$$ | ||
| + | |||
| + | b) Let's assume that the field induction is directed inside the drawing. Then the resulting current is shown in the figure: | ||
| + |  | ||
| + | |||
| + | The figure also shows the charge signs on the capacitor plates and the direction of the field in them (over /against the current). | ||
| + | |||
| + | Let's write down the sum of the voltage drops for the left and right semicircles: | ||
| + | |||
| + | $$U_1 + U_3 = \frac{\varphi}{2}$$ | ||
| + | |||
| + | $$U_3 - U_2 = - \frac{\varphi}{2}$$ | ||
| + | |||
| + | From here: | ||
| + | |||
| + | $$U_1 = \frac{\varphi}{2} - U_3$$ | ||
| + | $$U_2 = U_3 + \frac{\varphi}{2}$$ | ||
| + | |||
| + | Nodes a and b are neutral, so the algebraic sum of the charges on the plates connected to the nodes is zero.: | ||
| + | |||
| + | $$q_1 - q_2 - q_3 = 0$$ | ||
| + | |||
| + | $$C_1 U_1 - C_2 U_2 = C_3 U_3$$ | ||
| + | |||
| + | Let's substitute our relations for the stresses: | ||
| + | |||
| + | $$C_1\frac{\varphi}{2} - C_1 U_3 - C_2 U_3 - C_2 \frac{\varphi}{2} = C_3 U_3$$ | ||
| + | |||
| + | $$U_3 = \frac{C_1 - C_2}{C_1 + C_2 + C_3}\frac{\varphi}{2}$$ | ||
| + | |||
| + | $$q_3 = \frac{C_3(C_1 - C_2)}{C_1 + C_2 + C_3}\frac{\varphi}{2}$$ | ||
| + | |||
| + | Savchenko apparently chose the opposite direction of induction, so the answer turned out to be a minus. | ||
| + | |||
| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $11.2.8.$ [Insert the problem statement] | |||
| ### Solution | |||
| a) Well, everything is pretty trivial here, the voltage on each capacitor: | |||
| $$U = \frac{\varphi}{2}$$ | |||
| So there is a charge on each capacitor: | |||
| $$q_1 = C_1 \frac{\varphi}{2}$$ | |||
| $$q_2 = C_2 \frac{\varphi}{2}$$ | |||
| b) Let's assume that the field induction is directed inside the drawing. Then the resulting current is shown in the figure: | |||
|  | |||
| The figure also shows the charge signs on the capacitor plates and the direction of the field in them (over /against the current). | |||
| Let's write down the sum of the voltage drops for the left and right semicircles: | |||
| $$U_1 + U_3 = \frac{\varphi}{2}$$ | |||
| $$U_3 - U_2 = - \frac{\varphi}{2}$$ | |||
| From here: | |||
| $$U_1 = \frac{\varphi}{2} - U_3$$ | |||
| $$U_2 = U_3 + \frac{\varphi}{2}$$ | |||
| Nodes a and b are neutral, so the algebraic sum of the charges on the plates connected to the nodes is zero.: | |||
| $$q_1 - q_2 - q_3 = 0$$ | |||
| $$C_1 U_1 - C_2 U_2 = C_3 U_3$$ | |||
| Let's substitute our relations for the stresses: | |||
| $$C_1\frac{\varphi}{2} - C_1 U_3 - C_2 U_3 - C_2 \frac{\varphi}{2} = C_3 U_3$$ | |||
| $$U_3 = \frac{C_1 - C_2}{C_1 + C_2 + C_3}\frac{\varphi}{2}$$ | |||
| $$q_3 = \frac{C_3(C_1 - C_2)}{C_1 + C_2 + C_3}\frac{\varphi}{2}$$ | |||
| Savchenko apparently chose the opposite direction of induction, so the answer turned out to be a minus. | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||