 Electrostatic energy of a three-plate capacitor

In the parallel-plate approximation, a central plate at potential $V$ between grounded plates at $z=\pm H$ forms two parallel capacitors of gaps $H+a$ and $H-a$. Summing their field energies gives the displayed formula. At fixed voltage it is minimized at $a=0$. Finite circular plates have fringe fields; the uniform-gap expression assumes those edge corrections are neglected.