For a parallel-plate capacitor at a fixed potential difference between the plates, which geometric or material factor primarily determines the electrostatic field strength between the plates?

Introduction:
Electrostatic field strength E between parallel plates is closely linked to the applied potential difference V and geometry. Understanding which factor controls E aids in designing capacitors and high-voltage insulation gaps.

Given Data / Assumptions:

• Uniform field approximation between large, closely spaced parallel plates.
• Fixed potential difference V (not fixed charge).
• Neglecting fringing effects.

Concept / Approach:
The ideal relation is E = V / d, where d is the plate separation. For a given V, reducing d increases E, while increasing d reduces E. Plate area and dielectric constant affect capacitance (C = ε * A / d) but do not directly change E at fixed V in the uniform-field model.

Step-by-Step Solution:
Start with field definition: E = V / dHold V fixed; E varies inversely with dArea A and dielectric permittivity ε influence charge stored Q = C * V but not E when V is fixedTherefore, the controlling factor for E (at fixed V) is the separation d

Verification / Alternative check:
From Gauss’s law in the parallel-plate approximation, surface charge density σ = ε * E. At fixed V, E is set by geometry (d), and σ adjusts accordingly. This corroborates E = V / d.

Why Other Options Are Wrong:

• Plate differential: Not a standard term; does not define E.
• Plate area: A affects capacitance, not E at fixed V.
• Dielectric constant only: ε affects C and σ, but E depends on V and d.
• Edge shape: Influences fringing, not the uniform-field magnitude.

Common Pitfalls:

• Confusing fixed-voltage with fixed-charge conditions.
• Assuming higher ε automatically increases field strength.

Final Answer:
Plate separation (distance)