The following text is from the book Concepts of Physics by Dr. H.C.Verma, from the chapter on "Capacitors", page 154, under the topic "Energy in the Electric Field in a Dielectric":
Consider a parallel-plate capacitor with a dielectric of dielectric constant $K$. The energy stored in the capacitor is $U=\frac 1 2 CV^2$. The energy density in the volume between the plates is
$$u=\frac{U}{Ad}=\frac{\frac 1 2 \left(\frac{K\epsilon_0A}{d}\right)V^2}{Ad}=\frac 1 2 K\epsilon_0\left(\frac V d\right)^2=\frac 1 2 K\epsilon_0E^2$$
where $E=V/d$ is the electric field between the plates.
We see that the energy density in the dielectrics is greater than that in vacuum for the same electric field. The dipole moments interact with each other so as to give this additional energy.
I understood the mathematical part of the above text, according to which, energy density in a dielectric ($u=\frac 1 2 K\epsilon_0E^2$) is greater than that in vacuum ($u=\frac 1 2 \epsilon_0E^2$) by a factor of $K$. A unit volume of dielectric stores an excess energy equal to $\frac 1 2 (K-1)\epsilon_0E^2$ other than what is stored by vacuum.
How is this excess energy in the same amount of volume stored? According to the author the reason is "The dipole moments interact with each other so as to give this additional energy", but I'm unable to understand this statement. I know dipole moment vectors in a dielectric try to align themselves with the external electric field applied. Thermal forces tend to destroy this kind of uniformity.
But, how do dipole moments interact to increase the energy density and where is the energy coming from? I believe in the Law of Conservation of Energy and so I think some other form of energy is converted to electrical potential energy. But why does this happen?
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This is true if we took an isolated capacitor and then put in the dielectric. i.e. if there was a fixed amount of charges on the plates, then yes the dielectric would thus decrease the electric field between the plates, lower the potential difference between the plates, and therefore increase the capacitance of the capacitor, as I'm sure you are familiar with.
However, if we are holding $Q$ constant then we really should be using the equation $U=\frac12\frac{Q^2}{C}$ which then predicts a decrease in energy as $C$ increases. What is going on? The issue is in what is being held constant. In your book's explanation $V$, and hence $E$ is being held constant. So then when you put in your dielectric more charge ends up on the plates. Loosely speaking, field lines start on positive charges and end on negative charges. Therefore, since we have the same electric field magnitude associated with more charges, we have a larger "field density", and hence a larger energy density.
Note that this is why the book says "for the same field" rather than "for any capacitor". The claim only holds true for comparing capacitors at the same potential difference. I think the given explanation about the dipoles is suspicious though, because the dipoles are present even in the scenario where we hold the charge constant and the energy density decreases. I would say in the constant potential case the additional energy comes from the battery, as more work is done to put more charges across the same potential difference.