Following are the two types of the
electrostatic voltmeter.

i)

**Quadrant type electrostatic voltmeter**which is used to measure voltage upto 10 kV to 20 kV.
ii)

**Attached disc type electrostatic voltmeter**which is used to measure voltages above 20 kV.__1.1Quadrant Type Electrostatic Voltmeter__

The instrument consists of four fixed metal
double quadrants arranged such that there is a small air gap between the
quadrants and the total assembly forms shallow circular box. Inside this box a
double sectored needle is suspended by means of a phosphor bronze thread. The
needle is suspended such that it is placed equidistant from above and below
quadrant plates as shown in the Fig 1.

As shown in the above Fig 1. the fixed quadrants
are connected together. The voltage to be measured either a.c. or d.c. is
connected between the fixed quadrants
and the moving needle. This needle rotates due to the electrostatic force set
up due to the charge accumulation on the quadrant plates. Then the suspension
exerts a controlling torque and the needle settles at the position where both
the torques, controlling and deflection, are equal.

There are two types of the electrical connections
in the quadrant electrometer,

i) Heterostatic connection

ii) Idiostatic connection

__1.2 Heterostatic Connection__

In this type of connection, a high voltage
battery is used to charge the needle to a voltage considerably higher than the
voltage to be measured . the connection diagram is as shown in the Fig 2.

In this connection, the quadrants are connected
together in diagonally opposite pairs. The moving van i.e. needle is positively
charged due to battery. The deflecting force due to top and bottom quadrants on
movable needle cancels each other on both sides. The only deflecting force
responsible is force of attraction between left quadrant and right moving
sector and force of repulsion between right quadrant and left moving sector.

__1.2.1Theory of Heterostatic Connection__

To obtain the torque equation for the
heterostatic connection, consider only one half portion of the needle with two
quadrant to it . this is shown in the Fig 3.

The needle is considered as a circle of circle
with radius r. now this arrangement of two quadrants with needle exactly in
between, resembles the two capacitors placed side by side. At equilibrium
position, as needle is placed symmetrically, the capacitances C

_{1}and C_{2}are equal. But when needle rotates , the value of one capacitor becomes greater tan other.
Let V

_{1}= Potential of needle, V = V voltage being measured
V

_{L}= Potential of quadrant L, V_{M}= Potential of quadrant M
Let the needle deflects in anticlockwise
direction through and angle θ.

C

_{1}_{}= Capacitance of right hand capacitor
C

_{2}= Capacitance of right hand capacitor
d = Distance of needle from either top or
bottom plates of quadrants

Note that needle spans through and angle (α+θ)
under quadrant M and half as there are two faces of vanes.

The two capacitances C

_{1 }are in parallel hence,
The total energy stored is,

LET T

_{θ}_{ }be the torque in the position θ then for an infinitesimal change d of the needle, the work done in moving system is T_{θ}_{ }dθ. This work done is equal to the increase in the stored energy dW.
Using the equation (7),

But the medium is air hence ε = ε

_{0},
The above expression is obtained considering
only two quadrants and half needle hence for all four quadrants the deflecting
torque will be doubled.

**Note**: The torque is positive only when 2V

_{1}(V

_{L}+V

_{M}). Now V is the potential to be measured and is equal to V

_{M}-V

_{L}

If potential of needle V

_{1}is very large compared to voltage to be measured then,**Note**: Thus heterostatic connection, the uniform scale is obtained.

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