A parallel plate capacitor

Question:

A parallel plate capacitor whose capacitance C is $14 \mathrm{pF}$ is charged by a battery to a potential difference $\mathrm{V}=12 \mathrm{~V}$ between its plates. The charging battery is now disconnected and a porcelin plate with $\mathrm{k}=7$ is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of ___________$\mathrm{pJ}$.

(Assume no friction)

Solution:

$\mathrm{U}_{\mathrm{i}}=\frac{1}{2} \times 14 \times 12 \times 12 \mathrm{pJ} \quad\left(\because \mathrm{U}=\frac{1}{2} \mathrm{CV}^{2}\right)$

$=1008 \mathrm{pJ}$

$\mathrm{U}_{\mathrm{f}}=\frac{1008}{7} \mathrm{pJ}=144 \mathrm{pJ} \quad\left(\because \mathrm{C}_{\mathrm{m}}=\mathrm{kC}_{0}\right)$

Mechanical energy $=\Delta \mathrm{U}$

$=1008-144$

$=864 \mathrm{pJ}$

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