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(B) The capacitance of a parallel plate capacitor is $C = \frac{A \epsilon_0}{d}$.The capacitance of a parallel plate capacitor with a dielectric is $

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$1$

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(B) The capacitance of a parallel plate capacitor is $C = \frac{A \epsilon_0}{d}$.The capacitance of a parallel plate capacitor with a dielectric is $C' = \frac{A k \epsilon_0}{d} = k C = 2C$.Plates $1$ and $2$ form a capacitor of capacitance $C$,plates $2$ and $3$ form a capacitor of capacitance $C' = 2C$,and plates $3$ and $4$ form a capacitor of capacitance $C$.Since plates $1$ and $4$ are connected,the system between $1$ and $3$ consists of the capacitor between $1-2$ and $2-3$ in series. Thus,$C_1 = \frac{C \cdot C'}{C + C'} = \frac{C \cdot 2C}{C + 2C} = \frac{2C}{3}$.The system between $2$ and $4$ consists of the capacitor between $2-3$ and $3-4$ in series. Thus,$C_2 = \frac{C' \cdot C}{C' + C} = \frac{2C \cdot C}{2C + C} = \frac{2C}{3}$.Therefore,the ratio $\frac{C_1}{C_2} = \frac{2C/3}{2C/3} = 1$.

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