Before You Begin

Question #42732

Difficulty: 8/10 | 🟢 Free

The Question

A carbon resistor has color bands: Red, Black, Red, Silver, indicating its resistance at a reference temperature $T_0 = 300 \, \text{K}$ . It is connected across a parallel combination of two non-ideal cells. Cell 1 has an EMF $E_1 = 12 \, \text{V}$ and Cell 2 has an EMF $E_2 = 6 \, \text{V}$ . At $T_0$ , the internal resistances of both cells are $r_{01} = r_{02} = 1000 \, \Omega$ .

According to Drude's model, the electrical resistance is inversely proportional to the relaxation time $\tau$ of the charge carriers. For the external resistor, $\tau$ varies with temperature $T$ as $\tau_R \propto T^{-2}$ . For the internal resistance material of the cells, the relaxation time varies as $\tau_r \propto T^{-1}$ .

If the entire circuit is operated at a steady temperature of $T = 600 \, \text{K}$ , what is the steady current flowing through the external resistor? (Assume the dimensions of the resistors and the EMFs of the cells remain strictly independent of temperature).

Options

A.
1.0 , \text{mA}
B.
1.8 , \text{mA}
C.
3.0 , \text{mA}
D.
3.6 , \text{mA}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Color Code Logic|Equivalent EMF for Cells|Relaxation Time Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the multistep linkage between macroscopic circuit analysis (color codes, equivalent EMF) and microscopic conduction theory (Drude's model).

The primary trap is treating the color-coded resistance and internal resistances as static values, ignoring their dependence on the relaxation time $\tau(T)$ .

Students often misapply Drude's model by either assuming $R \propto \tau$ instead of $R \propto 1/\tau$ , or by failing to update the internal resistances of the cells before applying Millman's theorem.

By forcing the recalculation of both $R$ and $r$ at a new thermal equilibrium, the problem penalizes rote application of Kirchhoff's laws without accounting for the dynamic, temperature-dependent nature of the components.

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