The figure shows a metal fuel tank in the shape of a rectangular prism: its base is a square with side a and height is h.

Sunday, 26 August 2018

1:18 AM

Machine generated alternative text: Question: The figure snows a metal fuel tank in the shape ot a rectangular prism: its base is a square witn side a and height is h. At a small distance d trom one side is a rectangular metal plate, dimensions a x 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 tact, d a ), so that the plate and the tank, when empty, constitute a simple parallel plate capacitor witn dimensions a by h and separation d, where edge effects may be neglected. Capillary ettects are also negligible. The second sketch snows the tank with a depth y otfuel, which has a dielectric constant K (Greek kappa, not k). Our plan is to use the capacitance beüeen plate and tankas a fuel gauge. (Hint: we can treat paäly filled capacitor as two capacitors.) d Tank h Fuel Figure 1 _ The left figure depicts an empty fuel tank. It has a square base, side a, and height h. A capacitor plate, size approximately a by h, is parallel to one wall, which is the other capacitor plate, at distance d away. Right figure depicts the fuel tank now paäially filled with fuel (yellow) up to a height y. This fills the capacitor to height y. part 1) In terms ot the variables named and any relevant constants, derive an expression tor the capacitance between the plate and the tank when the fuel reaches height y as shown. part 2) , where V is the volume of fuel in the tank. (Remember that d a so the volume of fuel in the capacitor is negligible compared Hence derive an expression for the sensitivity ot this fuel gauge, Witn that in the tank. part 3) Consider a tank witn side a — litre. Check 27 cm, and height h PF /litre 15 cm, with capacitance spacing d 1.0 mm and containing kerosene with K 1.8. Express the sensitivity of the gauge in picofarads per

 

Machine generated alternative text: Feedback tor Part 1) For a capacitor, C = ReoA/d so the lower part of the capacitor nas Clover — Reoay/d and the upper part has C upl»r = KEoay % (h + (K — l)y). Marks for this submissiom 0.00/0.33. This submission attracted a penalty of 0.11 Feedback tor Part 2) V = a2y, so dV = a2 dy and dy = dV/a2. So [Note that the units ot eo are pF/m, so this expression has the correct units.) Marks for this submissiom 0.00/0.33. This submission attracted a penalty of 0.11 y)/d. The two are in parallel, so their capacitances add, giving dC dy dy (K — 1) SO ad Feedback tor Part 3) dC dV 1 dC a2 dy da 8.85 PF m 1.00 x 10- mxa (in metres) 1000 litres x (1.8—1). For interest picofarads can be measured fairly easily, but one would nave to be careful about stray capacitance and the zero would need to be carefully adjusted. In practice, capacitative fuel gauges nave different geometries that are easier to install. Marks for this submissiom 0.00/0.33. This submission attracted a penalty of 0.11 A correct answer , which can be typed in as follows: ( (kappa-I ) *y+h) ) /d A correct answer is , which can be typed in as follows: A correct answer is 26, which can be typed in as follows:

 

 

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