Question

As is commonly known a car has a fuel level display pointer in the dashboard. Liquid level sensing in a tank utilizes a capacitor and the dielectric properties of the fuel to measure the capacitance. The fuel tank of your car is 75% full and the rest of the tank contains air.
a. Research online for to find the capacitance formula when a dielectric is between the plates.
b. If the voltage of the car battery is 12V and the charge on the capacitor is 4C. The total area of one of the plates is 4 x 10-2 m2 and the separation of the plates is 2cm. Find the dielectric constant of the fuel.

Answer 1

a. $C = kε_0 \frac{A}{d}$

b. The capacitance

$C = C_{air} + C_{fuel}$

$C = kε_0 \frac{A_{air}}{d} + kε_0 \frac{A_{fuel}}{d}$

$A_{air} = 4 \times 10^{-2} \times 0.25 = 1 \times 10^{-2} m^2$

$A_{fuel} = 4 \times 10^{-2} \times 0.75 = 3 \times 10^{-2} m^2$

$C = \frac{ε_0 \times 10^{-2}}{d}[1 + 3k]$

$C = \frac{8.854 \times 10^{-12} \times 10^{-2}}{0.02}[1 + 3k]$

$C = 4.427 \times 10^{-12}[1 + 3k]$

Charge = 4C

Capacitance C = Q/V

$C = \frac{4C}{12} = \frac{C}{3}$

$\frac{C}{3} = 4.427 \times 10^{-12}[1 + 3k]$

$\frac{C}{3 \times 4.427 \times 10^{-12}} = 1 + 3k$

$3k = \frac{C}{1.328 \times 10^{-11}} – 1$

$k = \frac{[\frac{C}{1.328 \times 10^{-11}} – 1]}{3}$