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Question: The ligure shows metal fuel lank in lhe shape of rectangular prism: its base is square wilh side and height is h. At small dislance d from one side is rec...

Question

Question: The ligure shows metal fuel lank in lhe shape of rectangular prism: its base is square wilh side and height is h. At small dislance d from one side is reclangular metal plate, dimensions h. It is red in the sketch and is almost touching the floor and the walls of the tank The sketches are not to scale (in fact; d < < so that the plate and the tank when empty; constitute simple parallel plate capacitor with dimensions by h and separation d, where edge effects may be neglected. Cap

Question: The ligure shows metal fuel lank in lhe shape of rectangular prism: its base is square wilh side and height is h. At small dislance d from one side is reclangular metal plate, dimensions h. It is red in the sketch and is almost touching the floor and the walls of the tank The sketches are not to scale (in fact; d < < so that the plate and the tank when empty; constitute simple parallel plate capacitor with dimensions by h and separation d, where edge effects may be neglected. Capillary effects are also negligible: The second sketch shows the tank with depth y of fuel which nas dielectric constant (Greek kappa not k); Our plan is to use the capacitance between plate and tank as fuel gauge . (Hint we can treat partly filled capacitor as two capacitors. Tank Fuel Figure The eft figure depicts an empty fuel tank: Ithas square base, side and height h capacitor plate size approximately by h, is parallel t0 one wall, which is the other capacitor plale, at distance away: Right figure depicts the fuel tank now partially filled with fuel (yellow) up to height y: This Ills Ihe capacitor to height y: Part 1) In terms of the variables named and any relevant constants derive an expression for the capacitance between the plate and the tank when the fuel reaches height y as shown Part 2) Hence derive an expression for the sensitivity of this fuel gauge, capacitor IS negligible compared with that in the tank where V is the volume of fuel in tne tank (Remember that d < < So the volume of fuel in the Part 3) Consider tank with side a = 63 cm; ad height h = 15 cm_ with capacitance spacing d = 1.0 mm and containing korosene with k 1.8. Express tho sensitivity of the gauge in picofarads per litre dv pF /litre



Answers

A Volume Gauge. A tank containing a liquid has turns of wire wrapped around it, causing it to act like an inductor. The hiquid content of the tank can be measured by using its inductance to determine the height of the liquid in the tank. The inductance of the tank. The from a value of $L_{0}$ corresponding to a relative permeability of 1 when the tank is empty to a value of $L_{f}$ corresponding to a relative permeability of $K_{m}(\text { the relative permeability of the liquid) }$ when the tank is full. The appropriate electronic circuitry can determine the inductance to five significant figures and thus the
effective relative permeability of the combined air and liquid within the rectangular cavity of the tank. The four sides of the tank each have width $W$ and height $D$ (Fig. 30.34$)$ . The height of the
liquid in the tank is $d$ . You can ignore any fringing effects and assume that the relative permeability of the material of which the tank is made can be ignored. (a) Derive an expression for $d$ as a function of $L$ , the inductance corresponding to a certain fluid height, $L_{0}, L_{f},$ and $D .$ (b) What is the inductance (to five significant figures) for a $\operatorname{tank} \frac{1}{4}$ full, $\frac{1}{2}$ full, $\frac{3}{4}$ full, and completely full if the tank contains liquid oxygen? Take $L_{0}=0.63000 \mathrm{H}$ . The magnetic susceptibility of liquid oxygen is $\chi_{m}=1.52 \times 10^{-3} .$ (c) Repeat part (b) for mercury. The magnetic susceptibility of mercury is given in Table $28.1 .$ (d) For which material is this volume gauge more practical?

Okay, so the total magnetic flux can be equal to fire Airbus by and fires you go to be Europe attends air and plus and five equals is equals will be liquid times a local beers the magnetic If you at the airport and Ares is the area airports and Bill liquid is the magnetic few l'equippe are a logo Is the area adequate bar? So, you know, be liquid A liquid can be go to km user on my over w tested up came here is the road safety I'm sorry Relative probability over the liquid Where from the museum Museo is a probability constant And I was the current earned Eastern on returns Yeah w here is the with off the tank and d here is the high off the liquid. So as you can tell, um w can we cancel and this will give us museum on eye candy is able to be liquid a liquid and beer air is good museum on Iow testy minus the SW Since we know air is it was a team honesty, uh has w as you gets out that we can we can sew up and he's with us Beer air is with Museo at theme honesty. So there will have the total magnetic flux people The mu zero I came D plus museum on a D minus t which would give us a museum. Items came D plus capital D minus lower case T. We will have outwards to the inductive is the goto environ by which will eventually give us use your eye square does Kim deep rusty modesty. So if we expend it will have allergies for the museum and square km D plus easier and square times capital D and M minus Build your own square lower case D so that al zero school equals a museum and square capital d Our basic with museum and square km lower case T This will give us how is good at zero minus Algeria. Oh, Lord, is the over capital D last out after lower case t over capital D would you give us the high off the liquid which the lower case t here is? You go to Al Mines Algeria over LF minds out zero times Capital D So for a second question, we know how can we go out zero times one plus x m has. OK, see over a capital D tax em. Here is the magnetic susceptibility off the nigari oxygen. Okay, so we know Al zero is 0.63 Henry and accent is you One point by two constant warning. Three. So there will have allies with Joe 0.63 Henry Test wall plus 1.52 attesting to party three as lower case t over capital D. So we know when de over these you on for we'll have our Is he good with 0.630 point. I'm sorry. 0.630 to 9, Henry. And when the over these would want half have always with 0.6304 a m and when d already is equal to 3/4 hours will be able to on 0.63372 Harry. So when the o. K. C over Capital Diesel, the one who have always will do Joe 10.3096 Harry for next question. So for Mercury, we know how zero sent, which is 0.3 Harry by the magnetic susceptibility is different. So they x m for the mercury is negative. 2.9 tensed into party five you re plugging will have its such unquestionable l. So what the over the years with one? For we'll have our physical, the job 0.6 threes or the zero age. And when d o d 0/2 When I was able to draw a 00.629 and nine Henry. And with the over these will, three or four who have the inductions for the liquid mercury is throw 40.6 to 9 and Henry And when the over the top of the one who has the doctors political mercury, which is out here is you got a 0.6 to 9 a. Henry. So for the last person, while my answer is, uh, poop, it should be oxygen gristle, the liquid oxygen. It's more practical. Okay, It's because there is an easily detectable spread off various for the liquid oxygen. But this is not a case for the mercury. Okay, so there. Well, the answer should be the liquid oxygen is more practical. Okay, And these are the answers for this question.

Everybody. So we need to find a dry if the expression of K efficient as function age. And so for a we have C one is gonna be cake epsilon not let me get the absolute not to look nice, Absalon Not and, oh times, um over de and we have w each And now we have C to which equals up salon, not w out minus age de. And now we do see equation equals C one plus c two equals k absolute not w h de applies absolute night w l minus h over d and we can have equal to absolute not w d um k h minus. Plus l l minus h um, And then we confer this have absalon w l de equals one plus k h over al minus h o. Okay, And that is your answer for a And now for B, we just have to find the k efficient. So we have the k efficient equals. Um ah. And this is a l um, went h equals l or Okay, so, wine plus 1.9 five guy times I'll divided by four over l. Okay, and we're just using, um so for this above equation. When is asking about the k efficient and was gonna race this? This is the K efficient and so okay, efficient. Okay. And so we have minus l divided by four over Al. And we get wine plus 1.95 divided by four minus +14 And we get 1.24 OK? And we're not done. You start a little bit more to Dio. So now we have the k efficient at age equals l over half. So we have wine. Plus 1.95 times l two over l. Okay. Divided by al minus al divided by two divided by l equals one plus 1.95 divided by two minus 1/2. Okay, make that tool. But nicer car. And this is going to eagle 1.48 Now, um, the next thing next thing we're looking at is the k efficient at age equals 30 course. So course we have k efficient equals one plus 1.95 times three l or divided by l minus three l four divided by owl. And this whole thing was to get 1.71 Okay, And now for C. Um, we can solve, um, again the k efficient. So and this time is gonna be different because you're okay, efficient when the wind plus 33 instead of 1.95 and it's just gonna be the same as above. So we start with l or divided by owl, OK, minus l four. Divided by how? Okay. And this is unequal. Nine. Now, when the next one Which is gonna be one how it one plus 33 l half divided by l minus out half divided by l. And this is gonna equal to 17. Now we have deficient of the 3/4 a wine Quest 33 divided by three l fours. OK, divided by l, um, minus 304 divided by O by. And this is gonna equal Teoh 25 And de, we have, um, asking us, um, about for which fuel is a fuel gauge more practical and is, um, from B and C, it is clear, Say it is clear the methanol, the meth and all is the bust fuel among the given Teoh use and, um, in the series and because, um, it's groups of the sensors area sensors because of its Because it's wide range. Uh, die. We'll put dye electric, Constance, then gasoline. Okay, you guys. Thank you.

Mhm. And this problem? We have a parallel plate capacitor. Basically, um, used to measure the level of fuel in ah, fuel tank. So the idea here is that you have these two parallel plates on, then the at some, you know, they have some length here. L and then partially. They're partially submerged in the fuel, um too. And so h the distance eight is part is submerged in the fuel. And so because their fuel has a different guy electric constant than the air, we can figure out if we measure the capacities of this total, um, system here, we can figure out that we can figure out that the how what this fuel level would be. So what they want us to do is they want us to find the effective or total capacities as a function of h so h so again we can look at this is to parallel plate capacitor in, um, in parallel, just like we did in previous problem. And let's see, here we have the if their unparalleled, then the capacities is add up. So we have the capacity of the the air gap capacitor. Um, it's just the prime material free space area of the, um, uh, that has air between it divided by the distance between these two. Um, and this is the distance between it is the same for both cases. The area is eso will say that we have a wit w here, and that's going to drop out. So the area is l minus age times W So we get the capacitance of this part of the capacitor is that primitive ity of free space times l minus h times w divided by D And now for the part that is emerged in the fuel, we have the dialectic constant of the fuel. Um, the area cross sectional area of the plate. The, um that submerged in the fuel, the distant diet, productivity free space, and then the distance, which is also d. So a f is just w times h. And so now we can add the if we if we look at the total so we want to say the total capacity, Um, that's the part Nativity of the dialect of the total. Whatever the net material is in here, which is the effective compact, Um uh, effective dialectic constant, um, of the the relative, Um, the productivity of the, um of the material in here. And so now we get, um, the area is the total area, right? And then we have the distance is the same across everything. So this is the total capacity. And so what we want is we want to get this total, um, the total dialectic constant or the effective dialectic constant of the of this total capacity. So we know that that this must equal this plus this. So putting that all together and doing some rearranging and canceling some things out, we see the W's cancel out that these cancel out productivity of free space cancels out. And so we get the effective capacity could affect the dialectic constant of this system. Here is K, um, the dialectic constant of the fuel minus one. Um, the dialectic constant of the hair finds H over l plus one. And so then they tell us that in one case we have gasoline and one case we have methanol, right? And then we wanna look at the how the capacity this effective, um, die electric constant changes as the level of fuel changes. So gasoline has a dialectic constant of 1.95 which is not not a whole lot is about twice that. Of what areas. So we see here that are effective are basically the slope here which would give us the sensitivity How sensitive this is the changes in this. So this slope is about one in the first case. So at, um, when h is zero, obviously the effective di electric constant is one just the same as free space. When h s l. Then this is one and this cancels with that and we get the effective dialectic. Constant is just that of the fuel because it's completely it's emerged in the fuel. So we get one when h zero when h is h over ls also once a quarter of the way up, we get 1.24 When it's half the way up, we get 1.47 When it's three quarters of the way up, we get 1.72 And when it's all the way up, you get 1.95 So we go from 1 to 2, basically with close to one from 1 to 2. Um, and now what if we if we had methanol as our fuel. Nothing All has a very high dialectic constant. Well, I mean, relative to a gasoline anyway s so we could see that the sensitivity here is gonna be Instead of being one, it's gonna be 32. So it's gonna This is going to change much more, uh, much more rapidly. Is age changes so again, when ages zero, Um, Katie is one. And now when h over l is ah, quarter, we get Katie is nine. When it's a half, we get 17, and when it's three quarters, we get 25 then when h is l, then we get again. Um, Katie would be 33. So this would be much, much easier to sense. Um, I if we had method on here because the dialectic class and is changing a much greater amount as it goes from empty to full on DSO. We wouldn't have to be a sensitive to these changes measuring these changes to figure out how much fuel is in there with gasoline. We have to have much more kind of more sensitive circuitry to measure this dialectic constant because it's not changing Ah, whole lot. But again we could do it. I'm not sure if this is how fuel gauges work. I would doubt it, um, in a vehicle anyway, because, you know, this would be a pretty large parallel plates in there, but maybe I'm not sure how they how in a typical automobile, how the fuel level was actually measured. There's plenty of ways of doing it. Just not sure what was the most practical way and also probably the cheapest ous faras the manufacturers were concerned.

Hi, everyone as cylinder figure. The effective tapestries would be She went to the chateau. She even having the widow. Okay, absolutely not. The blue is the west of the plate and matches the height upon the and that's absolutely not w and minus edge upon the if effective gap distance of the effective direct replacement is K effective into absolute, not Omega. And upon day, so from here, effective you will get came by next month. Yeah, age upon a policeman. This is a party in part week value of K is given in the problem. 1.95 So I urge upon l is called to We went by food, so Okay, effective. You will get 1.95 minus bun into one upon four possible. So it will be 1.2375 Yeah, five. And upon l is going to have Okay, effectively 1.95 minus one. And to have plus one. So that is 1.47 fight. And for edge upon air is going to go through my foot. Okay. Effective. You will get 1.95 minus one into three by four plus pounds. So it is to be 12.7 point 25 sheep are. If he is 33 then fun edge upon end. It's got to burn upon phone. Okay, Effective building 33. My husband. And to win by four Lisbon, That is nine. What? At a pawn. And it's hard to have. Okay, effectively. Okay. Three minus pond into half. Plus one. 30 17 for had upon an escort to three by food. Okay, effectively, we 33 my husband into to buy food. And this one is 25. Yes, D This is a particular approach. Mhm. Yeah, that son thanks very much.


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