Chapter 25, Problem 24P

(II) Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire with undeformed length ℓ₀, cross-sectional area A₀, and resistance R₀. This sensor is rigidly affixed to the object’s surface, aligning its length in the direction in which length changes are to be measured. As the object deforms, the length of the wire sensor changes by ∆ℓ, and the resulting change ∆R in the sensor’s resistance is measured. Assuming that as the solid wire is deformed to a length ℓ, its density (and volume) remains constant (only approximately valid), show that the strain (= ∆ℓ/ℓ₀) of the wire sensor, and thus of the object to which it is attached, is ∆R/2R₀.