Missmatch in electrostatic force calc. by different methods

[JerryR]

I have been looking into the forces exerted by electrostatic fields and have come up with different answers using two different methods. I would appreciate any help in pointing to a reference that will reconcile this or point to an error in my methods. To keep things simple I am using a one dimensional model of a capacitor with a layered dielectric as shown in the figure below. The forces are being computed on a unit area basis.

http://C:\Users\Jerry\Documents\LaTeX\Force\Capacitor.png

(I hope the image embeds properly. The help screens were somewhat fuzzy on this)

Many references state that the force on the upper electrode will by given by $F_u = -\frac{1}{2}\epsilon_0 E_1^2$. It is not clear if the permitivity is always to be the permitivity of free space or if this is the material assumed to be near the electrode. In a similar manner the force on the lower electrode will be $F_l = \frac{1}{2}\epsilon_0 E_2^2$. A reference for the force on the bound charges at the dielectric interface (see below) shows that this force will be $F_i = \frac{1}{2}\epsilon_0 (E_1^2-E_2^2)$. As expected the sum of these forces equals zero.

Now for the alternate method. First compute the total energy stored in the capacitor. Next compute the change in the energy with a change of the $h_1$ and $h_2$ dimensions. This gives $F_u = -\frac{1}{2}\epsilon_1 E_1^2$, $F_l = \frac{1}{2}\epsilon_2 E_2^2$, and $F_i = \frac{1}{2}(\epsilon_1 E_1^2-\epsilon_2 E_2^2)$. Once again the sum of the forces is zero. However, the magnitudes are significantly different.

I would appreciate any guidance as to reconciling this difference or pointing to an error in my methods.

Referece = http://phys.columbia.edu/~nicolis/Surface_Force.pdf

 

[JerryR]

The image did not load. This basically shows two electrodes separated by a layered dielectric. The upper dielectric is $h_1$ thick with a permitivity of $\epsilon_1$. The electric field in this material is $E_1$. Similar for the lower material with the subscript of 2.

Could someone point me to a reference for including .png files in a post.

Thanks - Jerry