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A fuel gauge uses a capacitor to determine the height of the fuel in a tank. The effective dielectric constantKeff changes from a value of 1 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 of wand a length of L(Fig.). The height of the fuel between the plates is h. You can ignore any fringing effects.

(a) Derive an expression forKeff as a function of h.

(b) What is the effective dielectric constant for a tank 14 full, 12 full, and 34 full if the fuel is gasoline 1K= 1.952?

(c) Repeat part (b) for methanol

Short Answer

Expert verified

a)Keff=hLk-1+1

b) The effective dielectric constant for a tank is 14 full, 12 full, and 34 full if the fuel is gasoline Keff=1.24, Keff=1.47, Keff=1.71 respectively.

c) The effective dielectric constant for a tank is 14 full, 12 full, and 34 full if the fuel is methanol Keff=9, Keff=17, Keff=25 respectively

Step by step solution

01

) Derive an expression for Keff as a function of h.

The total capacity can be considered as two parallel capacitors, the first is the one of air and the second is the one of fuel so the total or effective capacity is given by

Ceff=Ca+Cf (1)

Air capacitance: We know that the capacitance of parallel plate capacitors, in general, is given by

Ca=0Aada (2)

Where

Aa=(L-h)wda=d

Substitution in (2) gives

C=oL-hd (3)

Fuel capacitance: The capacitance of a parallel plate capacitor filled with fuel, in general, is given by

Cf=Afdf (4)

Where:

Af=hwdf=d=K

Substitution in (4) gives

CJ=Kohwd (5)

Effective capacitance: Also effective capacitance is a capacitance of parallel plates so it is given by

Ceff=effAeffdeff

Where:

Aeff=Lwdeff=deff=Keff

Substitution in (5) gives

Ceff=Keff0Ld (6)

Step 4 Substitution in (1) gives

KeffoLwd=Kohwd+o(Lh)wd (7)

Simplify (7) to get that.

KeffL=Kh+(L-h)KeffL=h(K-1)+LKeff=h/L(K-1)+1

02

Calculate the effective dielectric constant or gasoline.

The effective constant for K = 1.95

when hL=14:

Keff14(1.95-1)+1=1.24

whenhL=12:

Keff12(1.95-1)+1=1.24=1.47

when hL=34:

localid="1668315039732" Keff34(1.95-1)+1=1.24=1.74

03

Calculate the effective dielectric constant or methanol.

The effective constant for K = 33.0.

when hL=14:

Keff14(33-1)+1=9

when hL=12:

Keff12(33-1)+1=17

when hL=34:

Keff14(33-1)+1=25

It's obvious that methanol is more practical than gasoline because we can easily change the capacity better than using gasoline

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