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Mechanical Energy and Force in a Capacitor
Figure 226 shows a simple twoplate capacitor, the plates of which are in a liquid or gaseous medium and are free to move with respect to each other. If a voltage v is applied to the plates of the capacitor a positive charge is distributed on one plate and an equal charge of opposite polarity is distributed on the other plate. These charges produce a force of attraction between the plates, and if the lefthand plate is fixed while the righthand plate is free to move, a force f is developed and tends to move the righthand plate toward the left. As a result electrical energy is converted to mechanical energy. However, the conversion of electrical energy to mechanical energy is generally accompanied by a change in the amount of energy stored in the dielectric. If the dielectric is lossless then all the energy absorbed by the dielectric is stored in reversible form and we have
where dW_{e} = differential energy supplied by the electrical source in time dtFor a fixed configuration of the dielectric, and in this case for a fixed position of the righthand plate relative to that of the lefthand plate, there can be no mechanical energy, and all of the energy supplied by the source is stored in the dielectric, i.e.
but
Substitution of Eq. 285 in 284 gives
It must be remembered that Eq. 286 is valid only for a fixed configuration of dielectric. The energy absorbed by the dielectric in changing the charge from q_{1} to q_{2} is
If the initial charge in the capacitor is zero, then the energy absorbed by the dielectric when the charge q is established is
In these equations the charge q is a function of the voltage across the capacitor expressed by Eq. 225 as q = vc. If the dielectric constant has a fixed value, then for a given configuration of the dielectric the capacitance C in Eq. 225 is constant, and the result of carrying out the integration in Eq. 288 is
Equation 289 expresses the energy that is stored in the dielectric of a lossless capacitor of capacitance C farads. This is true whether C is constant or variable. W_{st} is the stored energy for the particular value of C at a given instant regardless of what the value of C was prior to that instant or the value it might attain at some later instant. From Eq. 289 it is evident that a change in the applied voltage v or a change in the capacitance C produces a change in the stored energy. Hence, in general, the differential energy stored in the dielectric in time dt is expressed by
The differential energy supplied by the source is expressed by Eq. 286 whether the capacitance is fixed or variable, and if Eq. 225, 286, and 290 are substituted in Eq. 283 the result is
and the differential mechanical energy is
Equations 290 and 292 show that for a constant applied voltage (dv = 0) the differential mechanical energy equals the differential stored energy. Substitution of Eq. 225 in 292 yields
When Eq. 225 is substituted in Eq. 290 the result is
If q is held constant (this can be done by disconnecting the capacitor from the electrical source) the differential stored energy is expressed by
This is to be expected because there is no differential electrical energy input, and the differential mechanical energy must therefore result from a change in the stored energy. Equations 292 and 293 show that motor action results from an increase in the capacitance, which also follows from the fact that the coulomb forces act in such a direction as to increase the capacitance. The forces that are developed by an electric field in a dielectric are always in a direction that produces a dielectric configuration such that the capacitance becomes a maximum. The mechanical power developed during the differential time according to Eqs. 292 and 293 is
The force on the plates of the capacitor is found from
to be
If the arrangement is such that the developed forces produce rotation, the torque is expressed by
A practical application of converting the energy stored in a dielectric to electrical energy by means of mechanical energy input is in the use of the capacitor microphone, also known as the condenser or electrostatic microphone. In this application, sound waves cause motion of a diaphragm that is one plate of an electrically energized capacitor in which the other plate is fixed.


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