Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire that when undeformed has 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 Δℓ, and the resulting change ΔR in the sensor’s resistance is measured. Assuming that as the solid wire is deformed to a length ℓ, its density and volume remain constant (only approximately valid), show that the strain ( = Δℓ / ℓ₀ ) of the wire sensor, and thus of the object to which it is attached, is approximately ΔR / 2R₀.

Verified step by step guidance

1. Start by recalling the relationship between resistance and the physical properties of a wire. The resistance R of a wire is given by the formula: $R=\frac{\rho }{A}\ell$, where ρ is the resistivity, ℓ is the length, and A is the cross-sectional area of the wire.
2. Since the problem states that the volume and density of the wire remain constant during deformation, we can use the relationship for volume: $V=\ell ₀A₀\; =\ell A$. From this, we can express the new cross-sectional area A in terms of the original area A₀ and the change in length Δℓ: $A=\frac{A}{\ell }\ell$.
3. Substitute the expression for A into the resistance formula. The new resistance R becomes: $R=\frac{\rho }{A}\frac{\ell }{\ell }$. This shows that resistance depends on the square of the length of the wire.
4. Now, calculate the change in resistance ΔR due to the change in length Δℓ. Using the approximation for small changes, expand the square term: $R\approx R₀(1\; +\; 2\mathrm{\Delta \ell }/\ell ₀)$, where $R₀$ is the original resistance.
5. Finally, isolate the strain (Δℓ / ℓ₀) in terms of the change in resistance ΔR. From the previous step, we find: $\frac{\mathrm{\Delta \ell }}{\ell }\approx \frac{\mathrm{\Delta R}}{2}$. This shows that the strain is approximately equal to ΔR / 2R₀, as required.

Key Concepts

Strain

Strain is a measure of deformation representing the displacement between particles in a material body. It is defined as the change in length (Δℓ) divided by the original length (ℓ₀). Strain is a dimensionless quantity that indicates how much an object deforms under stress, and it is crucial for understanding how materials respond to external forces.

Resistance Change in Strain Gauges

In a strain gauge, the resistance of the wire changes as it deforms. This change in resistance (ΔR) is related to the strain experienced by the wire. The relationship between resistance and strain is derived from the physical properties of the wire, where the resistance is affected by changes in length and cross-sectional area, allowing for the measurement of strain through electrical resistance.

Gauge Factor

The gauge factor is a dimensionless number that quantifies the sensitivity of a strain gauge. It is defined as the ratio of relative change in electrical resistance to the mechanical strain. For a typical metallic strain gauge, the gauge factor is approximately 2, which means that a small strain will produce a proportional change in resistance, facilitating the measurement of strain in materials.