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...

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.)