A 20.0 ohm resistor has a rectangular cross-section with length, l, that is cons

Question

A 20.0 ohm resistor has a rectangular cross-section with length, l, that is constant and a width, w, that varies linearly from one end to the other of the resistor (that is dw/dx is constant). At one end, the rectangle is 3.40 cm wide, and at the other end, it is 8.55 cm wide. The long side of the rectangle has a width of 9.10 cm. The resistor is 8.45 cm deep. What is the resistivity of the material composing it? (Note: On the figure, side lengths are indicated, and the figure is not to scale. Assume that the shape has no effect on the resistance.)

Explanation / Answer

Consider the left diagram
from the left, at distance x < 8.45 cm, consider a small width dx
NOw, let the resistivity of material be r
then resistance of this segment = r*dx/A
where A = 9.1 cm * 2h [ where 2h is the verical height of the resistor at distance x from the left end]
from ratio proportion
3.4/m = 8.55/(8.45 + m) = 2h/(x + m)
28.73 + 3.4m = 8.55m
m = 5.5786

3.4/5.5786 = 2*h/(x + 5.5786)
h = (x + 5.5786)0.3047 = 0.3047x + 1.699

so, dR = r*dx/2*9.1*(0.3047x + 1.699)
integrating from x = 0 to x = 8.45
R = rln([0.3047*8.45/1.699 + 1]])/18.2*0.3047 = 0.166335r
but R = 20 ohm
r = 120.239 ohm cm = 1.202 ohm m

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