A fuel gauge uses a capacitor to determine the height of the...

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 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 w and a length $L\space (\textbf{Fig. P24.68}$). 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 $1\over4$ full, $1\over2$ full, and $3\over4$ 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?

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 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 w and a length $L\space (\textbf{Fig. P24.68}$). 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 $1\over4$
full, $1\over2$ full, and $3\over4$ 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?

Transcript

00:01 In this problem, we have a parallel plate capacitor basically used to measure the level of fuel in a fuel tank.

00:11 So the idea here is that you have these two parallel plates, and then they have some length here l.

00:20 And then partially, they're partially submerged in the fuel.

00:25 And so h, the distance h is submerged in the fuel.

00:29 And so because the fuel has a different dielectric constant than the air, we can figure out if we measure the capacitance of this total system here, we can figure out that we can figure out that the, how what this fuel level would be.

00:48 So what they want us to do is they want us to find the effective or total capacitance as a function of h.

00:53 So h.

00:55 So again, we can look at this as two parallel plate capacitors in parallel.

01:01 Just like we did in a previous problem.

01:05 And let's see here.

01:06 We have that if they're on parallel, then the capacitance is add up.

01:10 So we have the capacitance of the air gap capacitor.

01:15 It's just the permissivity of free space, the area of the, that has air between it, divided by the distance between these two.

01:27 And this is the distance between it is the same for both cases.

01:32 The area is, so we'll say that we have a width w here, and that's going to drop out.

01:40 So the area is l minus h times w.

01:42 So we get the capacitance of this part of the capacitor is that primitivity of free space times l minus h times w divided by d.

01:53 And now for the part that is submerged in the fuel, we have the dielectric constant of the fuel.

02:01 The cross -sectional area of the plate that's submerged in the fuel, the distance, permutivity free space, and then the distance, which is also d.

02:14 So af is just w times h.

02:19 And so now we can add the, if we look at the total, so we want to say the total capacity, that's the permutivity of the total, whatever, the net, material in here, which is the effective effective dielectric constant of the relative the permutivity of the material in here.

02:53 And so now we get the area is the total area, right? and then we have the distance is the same across everything.

03:02 So this is the total capacitance.

03:03 And so what we want is we want to get this total, the total dielectric constant or the effective dielectric constant of this total capacity.

03:15 So we know that this must equal this plus this.

03:21 So putting that all together and doing some rearranging and canceling some things out.

03:26 We see the ws cancel out, the ds cancel out, permutivity of free space cancels out.

03:32 And so we get the effective capacity, the effective dielectric constant of this system here is k the dielectric constant of the fuel minus one the dielectric constant of the air times h over l plus one and so then they tell us that in one case we have gasoline and one case we have methanol and then we want to look at the how the capacity this effective dielectric constant changes as the level of fuel changes so gasoline has a dialectic of 1 .95, which is not a whole lot, it's about twice that of what air is.

04:13 So we see here that our effective are basically the slope here, which would give us the sensitivity, how sensitive this is to changes in this...

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