A Capacitor Burglar Alarm. The capacitance of a capacitor

Chapter 26, Problem 90CP

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A Capacitor Burglar Alarm. The capacitance of a capacitor can be affected by dielectric material that, although not inside the capacitor, is near enough to the capacitor to be polarized by the fringing electric field that exists near a charged capacitor. This effect is usually of the order of picofarads (pF), but it can be used with appropriate electronic circuitry to detect a change in the dielectric material surrounding the capacitor. Such a dielectric material might be the human body, and the effect described above might be used in the design of a burglar alarm. Consider the simplified circuit shown in Fig. The voltage source has emf ??? = 1000 V. and the capacitor has capacitance ?C? = 10.0 pF. The electronic circuitry for detecting the current, represented as an ammeter in the diagram, has negligible resistance and is capable of detecting a current that persists at a level of at least 1.00 ???A for at least 200 ???s after the capacitance has changed abruptly from ?C? to C ? ?’. The burglar alarm is designed to be activated if the capacitance changes by 10%. (a) Determine the charge on the 10.0-pF capacitor when it is fully charged. (b) If the capacitor is fully charged before the intruder is detected, assuming that the time taken for the capacitance to change by 10% is short enough to be ignored, derive an equation that expresses the current through the resistor ?R as a function of the time ?t? since the capacitance has changed. (c) Determine the range of values of the resistance ?R? that will meet the design specifications of the burglar alarm. What happens if ?R? is too small? Too large? (?Hint?: You will not be able to solve this part analytically but must use numerical methods. Express ?R? as a logarithmic function of ?R? plus known quantities. Use a trial value of ?R? and calculate from the expression a new value. Continue to do this until the input and output values of ?R? agree to within three significant figures.) Figure: