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Question #77818

Difficulty: 8/10 | 🟢 Free

The Question

A galvanometer has a coil resistance $G = 99 \, \Omega$ and a full-scale deflection current $I_g$ at $20^\circ\text{C}$ . It is converted into an ammeter of range $I$ by connecting a shunt resistor of resistance $S = 1 \, \Omega$ at $20^\circ\text{C}$ . The temperature coefficients of resistance for the galvanometer coil and the shunt are $\alpha_G = 4.0 \times 10^{-3} \, ^\circ\text{C}^{-1}$ and $\alpha_S = 1.0 \times 10^{-3} \, ^\circ\text{C}^{-1}$ respectively. Assuming $I_g$ is independent of temperature, what is the relative rate of change of the ammeter's full-scale range with respect to temperature, $\frac{1}{I}\frac{dI}{dT}$ , at $20^\circ\text{C}$ ?

Options

A.
$2.97 \times 10^{-3} \, ^\circ\text{C}^{-1}$
B.
$3.00 \times 10^{-3} \, ^\circ\text{C}^{-1}$
C.
$3.96 \times 10^{-3} \, ^\circ\text{C}^{-1}$
D.
$4.95 \times 10^{-3} \, ^\circ\text{C}^{-1}$

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

Shunt Resistance Conversion|Temperature Coefficient (Alpha)|Relative Error Propagation in Powers

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to propagate systematic errors (temperature changes) through a non-linear function, distinguishing it from maximum random error propagation.

Students often fall into three traps: (1) blindly applying the maximum permissible error formula $\frac{\Delta I}{I} = \frac{G}{G+S}(\frac{\Delta G}{G} + \frac{\Delta S}{S})$ which incorrectly adds the coefficients, (2) approximating the range as $I \approx I_g \frac{G}{S}$ which leads to $\alpha_G - \alpha_S$ , or (3) assuming the shunt resistance is temperature-independent ( $\alpha_S = 0$ ).

The correct approach requires exact differentiation of the exact range formula, recognizing that temperature causes a correlated, directional shift in both resistors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Relative Error Propagation in Powers
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free