# calculate force on object between capacitor plates without knowledge of area

Electrical Engineering Asked by LeonTheProfessional on February 19, 2021

I am given an exercise with a very simple circuit

simulate this circuit – Schematic created using CircuitLab

I am asked to find the voltage of the voltage-source V1 such that an object with a given charge $$Q_{obj}$$ and mass $$m$$ of negligible volume lying between the plates of the capacitor C1 is hovering.

Equating a force with the gravitational acceleration and the mass of said object is no problem, I can figure that out, this is not what this question is about.

To find what force is acting on the object, I figured I want to find the electric field $$E$$ between the plates of the capacitor.

Now in all resources I could find, the area of the capacitor-plates is relevant for finding the electric field. However I am not given any area, just the distance between the plates $$d$$ is given. I think one usually assumes an infinite area in such case, but:
$$sigma = frac{Q_{cap}}{A}$$
and
$$E = frac{sigma}{varepsilon}$$
thus in this case there should be no electric field, and hence no force acting on my object?!

How can I find the electric field between the plates of a parallel-plate-capacitor from just the voltage $$V$$ and the distance between the plates $$d$$, without the area $$A$$?

In your current thought process, it looks like you're trying to find the charge on the capacitor (You denote it generically as $$Q$$, I'll denote it $$Q_{text{cap}}$$), which is proportional to the area of the plates. However, since we know that $$Q_{text{cap}}/C = V$$, you'll divide by the capacitance (which is itself proportional to the area), and your final result is area-independent.

A more straightforward formulation uses:

$$V = -int vec{E} cdot dvec{l}$$

Assuming the dielectic is uniform throughout the capacitor and edge effects are negligible (i.e. the particle is far from the edges), this turns into simple multiplication/division to relate the field to the capacitor voltage. You can subsequently relate the electric field and $$Q_{text{particle}}$$ to the electrostatic force on the particle that should hover:

$$vec{F}_{text{coul}} = Q_text{particle}vec{E}$$

and of course this should be equal in magnitude and opposite to the gravitational force on the particle.