# Missmatch in electrostatic force calc. by different methods

• JerryR
In summary, the conversation discusses the forces exerted by electrostatic fields in a one-dimensional model of a capacitor with a layered dielectric. The first method calculates the forces on the upper and lower electrodes using the permitivity of free space, while the second method calculates the forces using the permitivity of the respective materials. The sum of the forces is zero in both methods, but the magnitudes are different. The speaker is seeking guidance on reconciling this difference or identifying any errors in their methods. A reference is provided for the force on bound charges at the dielectric interface.

#### 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

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

## 1. What is electrostatic force mismatch?

Electrostatic force mismatch refers to the discrepancy between the calculated electrostatic force using different methods or equations. This can occur due to differences in assumptions, approximations, or numerical methods used in the calculations.

## 2. Why is electrostatic force mismatch important?

Electrostatic force plays a crucial role in many physical and chemical phenomena. Therefore, accurate calculations are necessary for understanding and predicting these processes. Any mismatch in the electrostatic force can lead to incorrect conclusions and affect the validity of the results.

## 3. How does electrostatic force mismatch occur?

Electrostatic force mismatch can occur due to several reasons, including the use of different equations or models to describe the same system, different numerical methods used for the calculations, or neglecting certain factors in the calculations.

## 4. How can electrostatic force mismatch be minimized?

To minimize electrostatic force mismatch, it is important to carefully choose the equations and numerical methods used in the calculations. Additionally, it is important to consider all relevant factors and minimize approximations in the calculations.

## 5. What are the potential consequences of electrostatic force mismatch?

Electrostatic force mismatch can lead to incorrect predictions and interpretations of physical and chemical processes, which can have significant consequences in various fields such as material science, biology, and engineering. It is therefore essential to minimize mismatch and ensure accurate calculations.