Question

As shown in the schematic below, a rod of uniform cross-sectional area $A$ and length $l$ is carrying a constant current $i$ through it and voltage across the rod is measured using an ideal voltmeter. The rod is stretched by the application of a force $F$. Which of the following graphs would show the variation in the voltage across the rod as function of the strain $\varepsilon $ when the strain is small. Neglect Joule heating.

• A.
• B.
• C.
• D.

Answer 1

Answer: (A)

When rod is stretched, its length increases.
Potential drop across the rod also increases due to increase in resistance of rod.
Resistance of rod
$R=\frac{\rho l}{A}=\frac{\rho l^{2}}{X}$
$ (\because$ volume of rod, $X = Al)$
Change in resistance of rod,
$\Delta R=\frac{\rho. 2l\Delta l}{X}$
Change in potential drop across rod,
$\Delta V=i \Delta R=\frac{i \cdot \rho \cdot 2 l \Delta l}{X}=\frac{2 i \rho l^{2}}{X} \cdot \frac{\Delta l}{l}=\frac{2 i \rho l^{2}}{X} \cdot \varepsilon$
or $\Delta V\propto\varepsilon$
As change in potential drop is directly proportional to strain,
voltage as a function of strain is as shown below;