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A capacitor consists of two stationaryplates shaped as a semi-circleof radius Radn a movableplate made of dielectricwith permittivity epsilon and capacble of rotoningaboutan axis O betweenthe stationary plates(fig). The thickness of the movables plateis equalto d which is practicallythe separationbetween teh stationaryplates. A potentialdifferenceV is appliedto the capacitor. FInd the magnitudeof the momentof forces relative to the axis O actingon teh movableplate in the position shown in the figure. |
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Answer» Solution :When the capacitor is kept at a contant potential difference `V`, the WORK performed by the moment of electrostatic forces between the plates when the inner moveable plate is rotatedby an angel`d varphi` equals the increasesin the potential energy of the system. This comes about because when chagres are made, chagres flow from the battery to keep the potentialconstant and the AMOUNTOF the work done by these charges is twice it magnitudeand opposite in sign to the changein the energy of the capacitor Thus `N_(z) = (del U)/(del varphi) = (1)/(2) V^(2) (del C)/(del varphi)` Now the capacitor can be thought of as made up two parts (with and without the dielectric) in parallel. Thus `C = (epsilon R^(2) varphi)/(2d) + (epsilon_(0) epsilon (pi - varphi) R^(2))/(2d)` as the area of a sector of angle `varphi` is `(1)/(2) R^(2) varphi`. Defferenctian then gives `N_(s) = - ((epsilon - 1) epsilon_(0) R^(2) V^(2))/(4d)` The negative sign of `N_(x)` indicates that the moment of the force is acting clockwire (i.e., trying to SUCK in the-dielectric). |
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