Capacitor plate spacing and dielectric breakdown

Dielectric breakdown occurs when the electric field in the insulator exceeds its dielectric strength $E_{br}$ (the maximum field it can withstand, e.g. about 3 kV/mm for air, 20–40 kV/mm for oil and polymers).

For a parallel-plate capacitor with voltage $V$ and plate spacing $d$, the field in the dielectric is

$$E = \frac{V}{d}$$

So for a given voltage, the field is inversely proportional to the spacing. The breakdown voltage is

$$V_{br} = E_{br}\,d$$

Why small spacing breaks down sooner

1. When $d$ is reduced, the same voltage produces a larger field, so $E$ reaches $E_{br}$ at a lower voltage.
2. A higher field gives the few free electrons more energy between collisions ($e E \lambda$), so they can ionize atoms and start an avalanche at a lower voltage.
3. A higher field also means more leakage current and dielectric heating per unit volume ($\propto E^2$), leading to thermal breakdown sooner.
4. In thin soft films, the electrostatic pressure $\frac{1}{2}\varepsilon E^2$ is larger and can cause electromechanical collapse.

Example

For air ($E_{br} \approx 3\ \text{kV/mm}$):

So a capacitor with closer plates fails at a much lower voltage. This is why high-voltage capacitors use either larger spacing or a dielectric with very high strength (mica, polypropylene), and each capacitor has a rated working voltage. (Very thin films often show a slightly higher $E_{br}$ because they contain fewer defects, but their breakdown voltage $E_{br}d$ is still lower.)