### Electrostatic instruments

Basically electrostatic instruments are all voltmeters. Fractically such instruments may be used for measurement of current and power but both the types of measurements require measurement of voltage across a known impedance. The main advantage of such instruments is the measurement  of high voltage in both a.c. and d.c. circuit without any errors  due to eddy current losses and hysteresis.
1.1 Principle of operation
The operation of all electrostatic instruments is based on the principle that there exists a force between the two plates with opposite charge. This force can be obtained using the principle that the mechanical work done is equal to the stored energy if there is a relative motion of plates.
Consider two plates A and B where plate A is fixed while B is movable. Two plates are oppositely charged and plate B is restrained by a spring  connected to fixed point. Let the force of attraction between the two plates be F newton. Let the capacitance between the two plates be C farad.

The energy stored E is the given by,

When applied voltage increases by dV, the current flowing through capacitance also changed and it is given by,

The input energy is given by,

Also due to change in applied voltage by value dV, the capacitance increases by Dc. Because plate B moves towards a fixed plate A which decreases the distance of separation between two plates increasing net capacitance.
Thus the new energy stored is given by,

The change in stored energy is given by,

Neglecting higher order terms of small quantities such as dC and dV, we can write,

From the principle of the conversation of energy, we can write input energy = increment in stored energy = increment in stored energy + mechanical work done.

From above expression it is clear that the force of attraction is directly proportional to the square of applied voltage V.
The above theory can be extended to the rotational motion, with the angular deflection θ  in place of the linear displacement x.

If the meter uses the spring control with torsional spring constant K than,

Such an instrument can be used for a.c. and d.c. measurements as the deflection is proportional to the square of the voltage to be measured. It shows square law response hence the scale is nonunifom which is compressed at the lower end.