Before You Begin

Question #43328

Difficulty: 8/10 | 🟢 Free

The Question

An insulated calorimeter of constant heat capacity $C$ contains a coil of constant resistance $R$ and a temperature-dependent inductance $L(T) = \frac{\alpha}{T}$ , where $T$ is the absolute temperature of the system and $\alpha$ is a positive constant. The coil is connected to an external AC voltage source of constant RMS value $V_{rms}$ and angular frequency $\omega$ . At the initial temperature $T_0$ , the inductive reactance of the coil is exactly equal to its resistance $R$ . Assuming all electrical energy dissipated by the coil is entirely converted into heat within the calorimeter, find the time $t$ required for the system's temperature to reach $2T_0$ .

Options

A.
$\frac{3 R C T_0}{2 V_{rms}^2}$
B.
$\frac{2 R C T_0}{V_{rms}^2}$
C.
$\frac{R C T_0}{V_{rms}^2}$
D.
$\frac{R C T_0}{2 V_{rms}^2}$

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

Inductive Reactance Dependency|Internal Energy Dependency|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple AC circuit theory with thermodynamics, specifically targeting the trap of ignoring the dynamic feedback loop between internal energy (temperature) and electrical impedance.

Students commonly misapply the power formula by assuming the inductive reactance $X_L$ remains constant at its initial value, leading to a constant power dissipation $P = V_{rms}^2 / 2R$ and an incorrect time $t = 2RCT_0/V_{rms}^2$ (Option B).

The rigorous approach requires recognizing that as the system heats up, the magnetic permeability drops, decreasing the inductance and impedance, which in turn increases the RMS current and the rate of Joule heating.

This creates a non-linear differential equation for temperature that must be integrated.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Inductive Reactance Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free