Single-axis torsional mirror

The electric field can be found from the gradient of the potential. In cylindrical coordinates, the gradient is given by 

The charge density on the plate on the counter-electrode is given by EqE, so that the total charge on the counter electrode is given by 

The torque on the mirror can be found from the principle of virtual work as described in Chapter 3, equation (3.6). Here, if the mirror angle increases a small amount Ad, the mechanical work done by the electrostatic actuator will be 

The force between the mirror and the counter-electrode is attractive, so that work is done by the mirror when the distance between them decreases. The work done by the mirror plus the work done by the battery must be equal to the change in the potential energy of the mirror. If the voltage on the actuator is held constant, as the mirror rotates toward the counter-electrode, the battery will have to supply charge AQ to keep the voltage constant as the capacitance increases, where AQ is give by 

Q = CV AQ = VAC. The battery must do work AW to supply this charge  

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The deflection angle 6q of a single-axis torsional mirror as a function of applied voltage V has been solved analytically by Senturia [7] in cylindrical coordinates, which we follow here. The mirror is grounded and a voltage V is applied to the counter-electrode. The mirror tilt angle 6 is positive in the clockwise direction (Figure 4.5). 

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