Before You Begin

Question #43356

Difficulty: 8/10 | 🟢 Free

The Question

A potentiometer wire of initial physical length $L_0$ , uniform cross-sectional area $A_0$ , and resistivity $\rho_0$ at ambient temperature $T_0$ is connected in series with an ideal constant current source $I$ . A standard cell of EMF $E_s$ is balanced across a physical length $l_0$ of the wire at $T_0$ . Due to Joule heating, the entire wire reaches a uniform steady-state temperature $T = T_0 + \Delta T$ . The wire material is isotropic with a linear coefficient of thermal expansion $\alpha$ and a temperature coefficient of resistivity $\alpha_r$ . Assuming $\alpha \Delta T \ll 1$ and $\alpha_r \Delta T \ll 1$ , what is the new physical balancing length $l$ for the same standard cell at the steady-state temperature $T$ ?

Options

A.
$l_0 [1 + (2\alpha - \alpha_r)\Delta T]$
B.
$l_0 [1 + (\alpha - \alpha_r)\Delta T]$
C.
$l_0 [1 + (\alpha_r - 2\alpha)\Delta T]$
D.
$l_0 [1 - \alpha_r \Delta T]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Potentiometer Sensitivity|Thermal Expansion of Area and Volume|Current Density Vector

Why This Question Is Tricky

Common Mistakes

This question tests the coupled effects of thermal expansion and temperature-dependent resistivity on Ohmic potential drops.

A common JEE trap is blindly applying the standard resistance formula $R(T) = R_0(1+\alpha_r \Delta T)$ .

This formula is an approximation that assumes geometric thermal expansion is negligible compared to resistivity changes.

However, in a potentiometer, the potential gradient depends on the local resistance per unit length, which is inversely proportional to the cross-sectional area.

By ignoring the area expansion $A(T) = A_0(1+2\alpha \Delta T)$ and only accounting for length expansion, students will incorrectly calculate the potential gradient and arrive at the trap option $l_0[1+(\alpha-\alpha_r)\Delta T]$ .

The problem forces students to derive the resistance from fundamental geometric and material properties at the new temperature.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Potentiometer Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free