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(III) The quantity of liquid (such as cryogenic liquid nitrogen) available in its storage tank is often monitored by a capacitive level sensor. This sensor is a vertically aligned cylindrical capacitor with outer and inner conductor radii Ra and Rb, whose length ℓ spans the height of the tank. When a nonconducting liquid fills the tank to a height h (≤ ℓ) from the tank’s bottom, the dielectric in the lower and upper region between the cylindrical conductors is the liquid (Kliq) and its vapor (KV), respectively (Fig. 24–33). (a) Determine a formula for the fraction F of the tank filled by liquid in terms of the level-sensor capacitance C. [Hint: Consider the sensor as a combination of two capacitors.] (b) By connecting a capacitance-measuring instrument to the level sensor, F can be monitored. Assume the sensor dimensions are ℓ = 2.0 m. Ra = 5.0 mm, and Rb = 4.5 mm. For liquid nitrogen (Kliq = 1.4, KV = 1.0), what values of C (in pF) will correspond to the lank being completely full and completely empty?
FIGURE 24–33 Problem 64.
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