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 sur- rounding 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 Figure. . The voltage source has anf €= 1000 V, and te capacitor has capacitance C = 10.0 pF. The electronic circuitry for detecting the current, represented as an ammeter in the dia- gram, has negligible resistance and is capable of detecting a cur- rent that persists at a level of at least 1.00 μA for at least 200 μs after the capacitance has changed abruptly from C to C’. The bur- glar 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 capaci- tance to change by 10% is short enough to be ignored, derive an equation that expresses the current through the resistor R as a func- tion of the time f 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 func- tion 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.)
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 sur-
rounding 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
Figure. . The voltage source has anf €= 1000 V, and te
capacitor has capacitance C = 10.0 pF. The electronic circuitry
for detecting the current, represented as an ammeter in the dia-
gram, has negligible resistance and is capable of detecting a cur-
rent that persists at a level of at least 1.00 μA for at least 200 μs
after the capacitance has changed abruptly from C to C’. The bur-
glar 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 capaci-
tance to change by 10% is short enough to be ignored, derive an
equation that expresses the current through the resistor R as a func-
tion of the time f 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 func-
tion 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.)
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