17) Two condensers have capacities \(C\) and \(4C\) respectively. If the charge on each condenser is doubled, the ratio of the energy stored in the condensers is:
Solution:
The energy stored in a capacitor is given by: \[ U = \frac{Q^2}{2C}. \]
When the charge is doubled, the new charge becomes \(2Q\). Therefore, the new energy stored is: \[ U' = \frac{(2Q)^2}{2C} = \frac{4Q^2}{2C} = \frac{2Q^2}{C}. \]
For the first condenser with capacitance \(C\): \[ U'_1 = \frac{2Q^2}{C}. \]
For the second condenser with capacitance \(4C\): \[ U'_2 = \frac{(2Q)^2}{2(4C)} = \frac{4Q^2}{8C} = \frac{Q^2}{2C}. \]
The ratio of the energies stored is: \[ \frac{U'_1}{U'_2} = \frac{\frac{2Q^2}{C}}{\frac{Q^2}{2C}} = \frac{2Q^2}{C} \times \frac{2C}{Q^2} = 4. \]
Hence, the ratio is \(4:1\) and the correct answer is option D.
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