But the math formula that you chose to use, does not in fact calculate the *actual* pressure between the plates. You are only calculating the *relative pressure* (compared to the outside plate pressure) between the plates because you have ignored the *outside positive pressure on the plates* by design!
The outside pressure is close to zero, which is why it doesn't get calculated. In simple terms, the change in energy is related to the fractional volume change. Between the plates, the fractional volume change is large for changes in plate separation. Outside the plates, the fractional volume change is many orders of magnitude smaller. Recall that pressure is
[latex]$P=-\frac{\partial E}{\partial V}$[/latex]
So since the changes in energy outside the plate are so much smaller than changes between the plates, the absolute value of the pressure outside the plates is many orders of magnitude smaller, and can be safely ignored. Which is why the effect is not observable for a single plate in the middle of a large cavity. It doesn't matter if the energy density outside the plates is larger, smaller, or whatever: it hardly changes with changes in plate separation, so its derivative is small and the pressure is close to zero. The pressure between the plates is large and negative.
You're hopelessly confused by your own math. You did *not* calculate the *actual* (total) pressure between the plates in that math formula. You calculated the *relative* pressure compared to the outside plate pressure.
Nope. It's the actual pressure, calculated using the expression I gave above. Energy in the vacuum between the plates increases with increasing separation, therefore pressure is negative. Quite simple, really. But yes, it takes a bit of math. Just like all real physics takes a bit of math: that's what
makes it physics and not philosophy.
The only issue here is that your math formula is a "gross oversimplification" of the actual physical processes at work in the chamber, and your calculation does *not* calculate the *actual pressure* in the chamber. It's a "relative to" sort of calculation at best case that has nothing to do with the *actual pressure* in the chamber.
Well, no. You are correct that the calculation is being performed "relative to" something else, but you're completely wrong about what it's relative to. It is
not calculated relative to the outside pressure (which is essentially zero). Rather, the energy between the plates is calculated for a displaced plate relative to an undisplaced plate. But both are energies inside the plate. From that calculation we can get the derivative of energy with respect to volume, a.k.a. pressure. But this is in fact an absolute pressure between the plates, not a relative pressure.
In that particular math formula, you specifically chose a *limited* and *relative* reference point for "pressure", nothing more.
Nope. A relative reference point for
energy is used, because when you calculate the derivative, only the change matters, you don't need an absolute energy, and you only need the derivative to calculate a pressure. But the pressure is still an absolute pressure, not a relative pressure. Your confusion betrays your ignorance of math.