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Question #42733

Difficulty: 8/10 | 🟢 Free

The Question

Four identical cylindrical resistors are connected to form a Wheatstone bridge $ABCD$ , with the resistors occupying arms $AB$ , $BC$ , $CD$ , and $DA$ . Each resistor has the color band sequence: Brown, Black, Brown, and Gold. A DC voltage source of $V = 50\text{ V}$ is connected across terminals $A$ and $C$ (current enters at $A$ and leaves at $C$ ). At a uniform ambient temperature of $20^\circ\text{C}$ , the bridge is perfectly balanced.

The resistor in arm $AB$ is then uniformly heated to $120^\circ\text{C}$ , while the other three resistors are maintained at $20^\circ\text{C}$ . For the resistor material, the temperature coefficient of resistivity is $\alpha_\rho = 4.0 \times 10^{-3}\ ^\circ\text{C}^{-1}$ and the coefficient of linear thermal expansion is $\alpha_L = 1.0 \times 10^{-3}\ ^\circ\text{C}^{-1}$ . Assuming isotropic thermal expansion and that the exact relations for temperature dependence apply (do not use small-parameter binomial approximations), calculate the magnitude of the open-circuit voltage across terminals $B$ and $D$ . Assume nominal resistance values for all calculations.

Options

A.
3.00\text{ V}
B.
4.17\text{ V}
C.
5.00\text{ V}
D.
2.27\text{ V}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Color Code Logic|Wheatstone Bridge Sensitivity|Thermal Expansion

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate material science (thermal expansion) with circuit theory, while avoiding two major traps.

First, students often memorize $\Delta R = R_0 \alpha \Delta T$ and assume $\alpha$ is solely the temperature coefficient of resistivity $\alpha_\rho$ , failing to account for the mechanical strain (geometric expansion) of the resistor where length increases by $(1+\alpha_L \Delta T)$ and area by $(1+\alpha_L \Delta T)^2$ .

Second, the large temperature change combined with unusually large coefficients makes the standard binomial approximation $(1+x)^{-1} \approx 1-x$ invalid, forcing the use of the exact rational function for resistance.

Distractor options correspond to ignoring geometric expansion entirely (4.17 V), adding the linear coefficient instead of subtracting (5.00 V), or subtracting the area expansion but forgetting the length expansion (2.27 V).

Quick Facts About This Question

Question Information

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