GATE 2022 IN Q.20: solution and answer

GATE 2022 · INQ.20 · 1 mark · MCQ

The resistance of a pure copper wire of length 10 cm and diameter 1 mm is to be measured. The most suitable method from amongst the choices given below is ________

(A)
Two wire method
(B)
Three wire method
(C)
Four wire method
(D)
Ellipsometry

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Answer and solution

ANSWER

The wire's resistance is only about $2.2 m Ω$ , a low resistance, so the four wire (Kelvin) method is the most suitable: option (C).

Official answer: CSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

First estimate the resistance to be measured, using $R = \frac{ρ L}{A}$ with copper resistivity $ρ \approx 1.7 \times 1 0^{- 8} Ω m$ , $L = 10 cm = 0.1 m$ and $d = 1 mm = 1 0^{- 3} m$ .
Cross-section: $A = \frac{π d ^{2}}{4} = \frac{π ( 1 0 ^{- 3} ) ^{2}}{4} \approx 7.85 \times 1 0^{- 7} m^{2}$ .
$R = \frac{1.7 \times 1 0 ^{- 8} \times 0.1}{7.85 \times 1 0 ^{- 7}} \approx 2.2 \times 1 0^{- 3} Ω = 2.2 m Ω$ This is a low resistance (well below $1 Ω$ ), so the method must keep lead and contact resistances out of the measurement.
Two-wire method: the same pair of leads carries the current and senses the voltage, so the reading is $R + R_{leads} + R_{contacts}$ . Lead and contact resistances are commonly several $m Ω$ to tens of $m Ω$ , comparable to or larger than the $2.2 m Ω$ being measured, so the error can be $100%$ or more.
Three-wire method: a lead-compensation connection used mainly with RTDs in bridge circuits. It cancels lead resistance only when the leads are matched, and it does not remove the contact resistance at the wire itself, so it is not suited to milliohm measurements.
Four-wire (Kelvin) method: one pair of current terminals drives the current $I$ through the wire. A separate pair of potential terminals, placed inside the current terminals, senses the voltage $V$ through a high-impedance voltmeter that draws almost no current. Lead and contact drops therefore do not appear in $V$ , and $R = V / I$ gives the wire's own resistance. This is the standard method for low resistances (the principle behind the Kelvin double bridge and four-terminal resistors).
Ellipsometry is an optical method: it measures the change in polarisation of light reflected from a surface to find thin-film thickness and refractive index. It does not measure the DC resistance of a wire.
So the four wire method, option (C), is the most suitable.

KEY IDEA

Work out the size of the resistance before picking a method. $R = ρ L / A \approx 2.2 m Ω$ is a low resistance (below $1 Ω$ ). At that size, lead and contact resistances are as large as the quantity being measured, so the method needs separate current and potential terminals: the four-terminal (Kelvin) connection.

WATCH OUT FOR

Picking the three wire method because it "compensates lead resistance". That scheme is meant for RTD bridges, depends on matched leads and does nothing about contact resistance at the specimen. Another mistake is skipping the estimate and treating this as an ordinary medium resistance, for which two-wire measurement would be good enough.

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 14 Bridges questions, with notes

GATE 2022 IN Q.20: solution and answer

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