Chapter 24, Problem 24.68P

A fuel gauge uses a capacitor to determine the height of the fuel in a tank. The effective dielectric constant K_eff changes from a value of K when the tank is empty to a value of K, the dielectric constant of the fuel, when the tank is full. The appropriate electronic circuitry can determine the effective dielectric constant of the combined air and fuel between the capacitor plates. Each of the two rectangular plates has a width w and a length L (Fig. P24.68). The height of the fuel between the plates is h. You can ignore any fringing effects. (a) Derive an expression for K_eff as a function of h. (b) What is the effective dielectric constant for a tank $\frac{1}{4}$full, $\frac{1}{2}$ full, and $\frac{3}{4}$ full if the fuel is gasoline (K = 1.95)? (c) Repeat part (b) for methanol (K = 33.0). (d) For which fuel is this fuel gauge more practical?

Figure P24.68