Resistance and resistivity: You have an unknown resistor (A). The resistor is de

Question

Resistance and resistivity: You have an unknown resistor (A). The resistor is depicted in the figure ow and has a cylindrical shape. You have the ability to measure the geometry of the resistor, and you measure the diameter to be can measure voltages. You measure the length of the resistor to be L, and you 2r What is the total resistance of resistor (A) in terms of its geometry? b) This unknown resistor is placed in series with a reference resistor R and a knowa voltage V'is applied across the two resistors as shown. You measure the voltage difference ar-p-v, between points a and b. What is this voltage difference AV in terms of , R, and R? Ree V-(Res +Ke (c) Determine the resistivity of the material in resistor (A) in terms of the resistance of resistor? ) (b)If the length of the unknown resistor is doubled, what is the resistivity of the material it is made of? 2el

Explanation / Answer

Answer: Length of resistor = L (given)

Diameter, d of resistor = 2r (given)

We know that the resistance, R is related to geometry by following relation:

RA = (L)/A; where, = Resistivity; L=Length; A=cross sectional area = d2/4 = (2r)2/4 = r2

Thus, RA = (L)/A = (L)/(r2) Answer (a)

When the unknown resistor is placed in series with the reference resistor, the same current, I will pass through the both resistors:

Thus, I = V/RA (for Unknown resistor)……………………………Eqn.(1)

Also, I = Vref/Rref (for reference resistor)……………………………..Eqn.(2)

Equating eqn(1) & (2), we have:

V/RA = Vref/Rref ……………………………………….Eqn.(3)

Also, V + Vref = V

Thus, Vref = V - V Substituting it in eqn.(3), we have:

V/RA = (V - V)/Rref

V/RA = V/Rref - V/Rref

V/RA + V/Rref = V/Rref

V*(1/RA + 1/Rref) = V/Rref

V = (V/Rref) / (1/RA + 1/Rref)

V = V / (Rref/RA + 1) or   VRA / (Rref + RA) Answer (b)

From Answer(a), RA = (L)/A = (L)/(r2)

Thus, resistivity = RA*(r2) / L    Answer (C)

Since resistivity is the material property, it remains unchanged upon changing the geometry of resistor (i.e. if length of the resistor is doubled.). Thus, resistivity will remain constant. Answer (d)

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