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F/A=-(d/da)(<E>/A)

"a" is the distance between the plates. So if the plates are moved closer together, the change in energy over area is negative, and the change in distance between the plates is negative. A negative number over a negative number is positive, times the minus sign before the derivative, giving a negative force. If the plates are moved apart, the change in energy is positive, over a positive change in distance, times the minus sign before the derivative, also resulting in a negative force.

But I thought the Casimir effect produces a conservative force? Is the Casimir effect always attractive (since it's always negative), or will moving the plates apart cause an additional push apart?

I know this is basic arithmetic, but my understanding of the physical effect doesn't seem to jive with the math. Any help would be greatly appreciated.