Some of you've participated in the Physics Challenge for Teachers and
Students.
Perhaps most of you agree with the solution provided by J. Iñiguez
from Universidad de Salamanca.
The "problem" lies with differentiating "1/2 CV squared" with respect
to x and claiming this to be the electric force. The "problem" is that
V also depends on x. In order to keep V constant while we change x
(and hence change the capacitance), we will have to change the amount
of charge stored in the capacitor, since C = Q/V. So to change x while
keeping V constant, we will also have to move the charges, and some of
the change in energy stored has to be used to do this.
The energy change in changing x does not completely go towards moving
the plates. It also goes partly to moving the charges. So
"differentiating "1/2 CV squared" with respect to x" is NOT equal to
the electric force between the plates.