calculate force on object between capacitor plates without knowledge of area

Question

I am given an exercise with a very simple circuit

simulate this circuit &ndash; 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 $A$?

nanofarad · Accepted Answer

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.

calculate force on object between capacitor plates without knowledge of area

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