26. Capacitors & Dielectrics

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 regions 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 tank being completely full and completely empty?

(II) Consider the circuit shown in Fig. 26–67, where all resistors have the same resistance R. At t = 0, with the capacitor C uncharged, the switch is closed. At t = ∞, the currents can be determined by analyzing a simpler, equivalent circuit. Identify this simpler circuit and implement it in finding the values of I₁, I₂ and I₃ at t = ∞.

Paper has a dielectric constant K = 3.7 and a dielectric strength of 15 x 10⁶ V/m. Suppose that a typical sheet of paper has a thickness of 0.11 mm. You make a “homemade” capacitor by placing a sheet of 21 cm x 14 cm paper between two aluminum foil sheets (Fig. 24–40) of the same size. What is the capacitance C₀ of your device?