Before You Begin

Question #43877

Difficulty: 8/10 | 🟢 Free

The Question

A series $L$ - $C$ - $R$ circuit consists of an ideal inductor of inductance $L$ , an ideal capacitor of capacitance $C$ , and a solid cylindrical resistor of initial length $l_0$ and cross-sectional area $A_0$ at $t=0$ . The resistor is made of a material with a constant electrical resistivity $\rho$ (strictly independent of temperature) and a coefficient of linear thermal expansion $\alpha$ . The circuit is placed in a thermal chamber where the ambient temperature increases linearly with time as $T(t) = T_0 + \beta t$ , where $\beta$ is a positive constant. At $t = 0$ , the capacitor is charged to $Q_0$ and the circuit is closed. The resistor remains purely ohmic throughout the process. Assuming the frequency of oscillation is extremely high ( $\frac{1}{\sqrt{LC}} \gg \alpha \beta$ ) and $\alpha \beta t \ll 1$ , which of the following expressions correctly represents the time evolution of the charge amplitude envelope $Q_{env}(t)$ on the capacitor? Let $R_0 = \frac{\rho l_0}{A_0}$ .

Options

A.
$Q_0 \exp\left[ -\frac{R_0}{2L} \left( t - \frac{1}{2}\alpha \beta t^2 \right) \right]$
B.
$Q_0 \exp\left[ -\frac{R_0}{2L} \left( t + \frac{1}{2}\alpha \beta t^2 \right) \right]$
C.
$Q_0 \exp\left[ -\frac{R_0}{2L} t \right]$
D.
$Q_0 \exp\left[ -\frac{R_0}{2L} \left( t - \alpha \beta t^2 \right) \right]$

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

Resistor Phase Invariance|L-C Circuit Analogy|Thermal Expansion of Area and Volume

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of thermodynamics and electrodynamics, specifically targeting the misconception that a resistor's thermal expansion inherently increases its resistance.

Students often blindly apply the standard temperature coefficient of resistivity formula $R = R_0(1 + \alpha \Delta T)$ .

However, when resistivity is constant, the geometric expansion dictates the resistance.

Because the cross-sectional area expands at twice the rate of the length ( $2\alpha$ vs $\alpha$ ), the overall resistance actually decreases as $R(t) \approx R_0(1 - \alpha \Delta T)$ .

Furthermore, the problem tests the application of a time-dependent damping factor $\gamma(t) = R(t)/2L$ in an L-C-R circuit, requiring students to integrate the decay rate rather than treating it as a constant.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Resistor Phase Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free