In Hypothesis #1, the
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Answer:
The work done on the gas is equal to the area under the curve pv diagram w = area of triangle = 1/2 (base)(height) = 1/2 (BC)(Ac) = 1/2 (3v - v)(3p - p) = 1/2 (9 vp - 3 vp - 3vp + vp) = 4 vp/2 W = 2 vp
Check attachment for the diagrammatic representation
The electrostatic energy stored in a capacitor with capacitance 
, with a voltage difference V applied to it, and without dielectric, is given by
Now let's assume we fill the space between the two plates of the capacitor with a dielectric with constant k. The new capacitance of the capacitor is
So, the energy stored now is
Therefore, the ratio between the energies stored in the capacitor before and after the introduction of the dielectric is
Now let's assume we fill the space between the two plates of the capacitor with a dielectric with constant k. The new capacitance of the capacitor is
So, the energy stored now is
Therefore, the ratio between the energies stored in the capacitor before and after the introduction of the dielectric is