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Homework Statement:

Is the external mechanical work done by inserting a dielectric material between two plates of a capacitor (plates capacitor) when the power source remains connected always negative?
and if so, does it mean that the process can proceed spontaneously after being initiated?
Relevant Equations:
 The original capacity is C0, the voltage is V0, and the dielectric constant of the material is K >1.
1. new capacity: ##C = K\cdot C_{0}##.
2. initial potential energy: ##U_{i} = \frac 1 2 \cdot C_{0}V_{0}^2##.
3. final potential energy: ##U_{f} = \frac 1 2 \cdot KC_{0}V_{0}^2##.
4. total work done: ##W_{tot} =\Delta U = U_{f}  U_{i} = \frac 1 2 \cdot (K1)C_{0}V_{0}^2 > 0##.
5. also total work done: ##W_{tot} = W_{p}~\text{(power source work for streaming more charge)} + W_{ext}~\text{(external work for inserting the material).}##
6. power source work: ##W_{p} = \Delta q \cdot V_{0} = (q_{f}  q_{i})V_{0} = (C \cdot V_{0}  C_{0} \cdot V_{0}) \cdot V_{0} = (K1) \cdot C_{0}V_{0}^2.##
7. external mechanical work: ##W_{ext} = W_{tot}  W_{p} = \frac 1 2 \cdot (K1) \cdot C_{0}V_{0}^2  (K1) \cdot C_{0}V_{0}^2 =
\\= \frac 1 2 \cdot (K1)C_{0}V_{0}^2 < 0.##
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