(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.
(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.
(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?
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