19. (III) Small distances are commonly measured capacitively. Consider an air-fi

Question

19. (III) Small distances are commonly measured capacitively. Consider an air-filled parallel-plate capacitor with fixed platearea A = 25 mm2 and a variable plate-separation distancex. Assume this capacitor is attached to acapacitance-measuring instrument which can measure capacitance C inthe range 1.0 pF to 1000.0 pF with an accuracy of C = 0.1 pF.(a) If C is measured while x is varied, over what range(xmin< x < xmax) can theplate-separation distance (in m) be determined by this setup?(b) Define x to be the accuracy (magnitude) to which x can bedetermined, and determine a formula for x. (c) Determine thepercent accuracy to which xmin and xmax canbe measured.

Explanation / Answer

C = 0 A /x       formula for a parallel plateair-filled capacitor x = 0 A / C xmin = 8.85 * 10E-12 * 25 * 10E-6 / (1000 * 10E-12)= .221 * 10E-6 = .221 m xmax = 1000 xmin since only thedenominator changes = 221 m dx = - 0 A / C2 dC dxmax = (1 / C) xmax dC dxmin = (1 / C) xmin dC Substitute the given values to calculate the level ofaccuracy and dxmax / xmax to get the fractionalaccuracy (multiplying by 100 to get %) dxmax = (1 / C) xmax dC dxmin = (1 / C) xmin dC Substitute the given values to calculate the level ofaccuracy and dxmax / xmax to get the fractionalaccuracy (multiplying by 100 to get %) dxmin = (1 / C) xmin dC Substitute the given values to calculate the level ofaccuracy and dxmax / xmax to get the fractionalaccuracy (multiplying by 100 to get %)

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