Electrostatic force calculations

1. MAEdwards

1
I am trying to calculate the forces in a system with a mix of conductive + dielectric components upon application of an AC potential. Frequencies are low enough that I need not consider magnetic field effects. I can solve equations to obtain the electric field, but at am a loss for force calculations. I am familiar with calculating the Maxwell stress tensor in purely capacitative systems. It is unclear to me whether I should be somehow incorporating conductivity as part of a complex dielectric constant and using this in the Maxwell stress tensor calculation or I should just use the regular dielectric constant.

If someone could suggest how I would go about this, even for a parallel plate system filled with a material with uniform conductivity and permittivity, I am sure I could carry it forward. I have looked through a number of basic textbooks, but find nothing to help me, but if anyone can refer me to one which answers my query I would be equally glad.

Many thanks,

Martin

2. stevenb

703
This is a good question and I agree that you don't often see this discussed in text books, hence I'm also not sure of the best answer. However, I can help brainstorm a little, just by restating things you already know.

Electric field is force per unit charge. Hence, if you solve an electro-static problem (really quasi-static because of AC fields) like this, you will have a solution for both electric field and charge density (really surface charge density). So, you should be able to integrate the net force over a surface using electric field and charge density.

The surface charge density is not constant in time because there is current flow with an AC potential. The analysis can be simplified with sinusoidal analysis which gives rise to the concept of complex permittivity (or complex conductivity). This comes from ampere's law, hence it is needed to calculate the surface charge density via surface current density.

Of course, none of these types of calculations are easy, but assuming you actually solve the field problem, then I'm thinking you shouldn't need to go back and use complex permittivity to calculate force. It seems you should be able to calculate a sinusoidally varying force from electric field and surface charge density directly. This should just be an integration operation. For example, with a parallel plate, there will be a force between the surface charge density in the conductor and that in the dielectric. This would be easy to calculate for large area plates.

Basically, the difference between DC and AC is that AC will generate a phase shift between the applied potential and the responding surface currents in the conductors. This results in delays in the build up of charge. However, whatever the charges are at a particular point in time, it is those charges that will exert force. Hence, it seems the field and charge solutions already contain the information you need, at least based on your simplifying assumptions.

The end result could show a force which is actually out of phase with the applied potential.