Chapter 24, Problem 64P

(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 R_a and R_b, 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 (K_liq) and its vapor (K_V), 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. R_a = 5.0 mm, and R_b = 4.5 mm. For liquid nitrogen (K_liq = 1.4, K_V = 1.0), what values of C (in pF) will correspond to the lank being completely full and completely empty?

FIGURE 24–33 Problem 64.