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(D) Let the capacitance of the air-cored capacitor be $C$. The capacitance of the capacitor with dielectric $K$ is $K C$.Initially,the charge on the a

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$\frac{V-V^{\prime}}{V^{\prime}}$

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(D) Let the capacitance of the air-cored capacitor be $C$. The capacitance of the capacitor with dielectric $K$ is $K C$.Initially,the charge on the air-cored capacitor is $Q = C V$ and the charge on the uncharged capacitor is $0$.When connected in parallel,the total charge is conserved and the capacitors reach a common potential $V^{\prime}$.The total charge $Q_{total} = C V + 0 = C V$.The total capacitance $C_{total} = C + K C = C(1 + K)$.Using the formula for common potential $V^{\prime} = \frac{Q_{total}}{C_{total}}$,we get:$V^{\prime} = \frac{C V}{C(1 + K)} = \frac{V}{1 + K}$.Rearranging for $K$:$1 + K = \frac{V}{V^{\prime}}$$K = \frac{V}{V^{\prime}} - 1 = \frac{V - V^{\prime}}{V^{\prime}}$.

An uncharged parallel plate capacitor having a dielectric of constant $K$ is connected to a similar air-cored parallel plate capacitor charged to a potential $V$. The two capacitors share charges and the common potential becomes $V^{\prime}$. The dielectric constant $K$ is

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