Sol. The work performed by the moment of forces M upon the rotation of the plate through an angle element dα is equal to the decrease in the electric energy of the system at q = const:

M Z dα = – dW | q ,

where W = q 2 /2C. Hence

M z = – = . .... (1)

In the case under consideration, C = C 1 + C e , and C e are the capacitances of the parts of the capacitor with and without the dielectric. The area of a sector with an angle α is determined as S = αR 2 /2, and hence

C = ε 0 αR 2 /2h + εε 0 (π – α) R 2 /2h.

Differentiating with respect to α, we find ∂ C/ ∂ α = (ε 0 R 2 /2h) (1 – ε). Substituting this expression into formula (1) and considering that C = q/U, we obtain

M z = (1 – ε)

= – (ε – 1) < 0.

The negative sign of M z indicates that the moment of the force is acting clockwise (oppositely to the positive direction of the angle α; see Fig.). This moment tends to pull the dielectric inside the capacitor.

It should be noted that M z is independent of the angle α. However, in equilibrium, when a = 0, the moment M z = 0.