# Work due to introducing a dieletric in a capacitor

I have a doubt about a couple of exercises tha asks to find the work done on introducing a dieletric between the plates of a capacitor. Yes, this question is in the general case, how do we procede? I really don't know how to start those questions. I know how to find the new capacitance, the new field between the plates and the new voltage, but I don't know how to find the work required to introduce the dieletric.

So, given a capacitor of capacitance $C$ and charge $Q$ what's the procedure if we want to find the work on introducing a dieletric of constant $k$ inside the plates?

I just want some ideas on how to start those problems.

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One way to compute the electrostatic energy stored into a plate capacitor is simply to calculate it as:

$E_{el} \equiv \int d^3r \: \frac{1}{2}\vec{D}\cdot\vec{E}$

which would be simply:

$E_{el} \equiv \int d^3r \: \frac{\epsilon_0}{2}\vec{E}^2$

in vacuum.

I would just consider the energy difference between before and after insertion of the dielectric medium

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