A simple fuel level gauge can use the dielectric constant of the fuel to indicat

Question

A simple fuel level gauge can use the dielectric constant of the fuel to indicate its level in a tank. Consider the schematic drawn in the figure where the two rectangular plates of the capacitor have a width w, and a length L. The height of the fuel between the plates is h. The effective dielectric constant, Keff, changes form a value of 1 when the tank is empty, to the value of the dielectric constant of the fuel, K, when the tank is full. The appropriate electronic circuitry can determine the effective dielectric constant of the system, a) Ignoring any fringe effects, derive an expression for the as a function of h. b) What is the dielectric constant for a tank that is 1 / 4 full. 1 / 2 full, and 3 / 4 full if the fuel is gasoline (K = 1.95)? c) Repeat part b for the case of methanol as the fuel, (K = 33.0). d) For which fuel is this a more appropriate gauge?

Explanation / Answer

a)

Ceff = C1 + C2

keff L w = 1 * (L-h) w + k * h w

keff = (L-h)/L+ k h/L

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part (b):

1/2 full:

keff = (L-h)/L+ k h/L = (1-1/2)/1+ 1.95* (1/2)/1 = 1.475

3/4 full:

keff = (L-h)/L+ k h/L = (1-3/4)/1+ 1.95* (3/4)/1 = 1.7125

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c)

1/2 full:

keff = (L-h)/L+ k h/L = (1-1/2)/1+ 33* (1/2)/1 = 17

3/4 full:

keff = (L-h)/L+ k h/L = (1-3/4)/1+ 33* (3/4)/1 = 25

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d)

methanol, because Keff is in a wide range for methanol.

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