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Question #43365

Difficulty: 8/10 | 🟢 Free

The Question

A real voltage source has an electromotive force $E = 8 \text{ V}$ . Its internal resistance depends on its temperature $T$ according to the relation $R_{\text{int}} = R_0 [1 + \alpha (T - T_0)]$ , where $T_0$ is the constant ambient temperature, $R_0 = 1 \ \Omega$ , and $\alpha = 0.01 \text{ K}^{-1}$ . The source dissipates heat to the surroundings according to Newton's Law of Cooling, with the rate of heat loss given by $P_{\text{cool}} = K(T - T_0)$ , where $K = 0.02 \text{ W/K}$ . The source is connected to a variable external load resistor $R_L$ (which is located far away and does not affect the thermal environment of the source). What value of $R_L$ maximizes the steady-state electrical power delivered to the load?

Options

A.
$6 \ \Omega$
B.
$1 \ \Omega$
C.
$2 \ \Omega$
D.
$\frac{1 + \sqrt{33}}{2} \ \Omega$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Maximum Power Transfer|Steady State Temperature Gradient|Internal Resistance Variation

Why This Question Is Tricky

Common Mistakes

This question tests the limits and fundamental assumptions of the Maximum Power Transfer Theorem.

Students typically memorize that maximum power is transferred when the load resistance equals the internal resistance ( $R_L = R_{\text{int}}$ ).

However, this theorem strictly assumes that $R_{\text{int}}$ is a fixed constant independent of the load.

In this coupled electro-thermal system, the internal resistance depends on the current, which in turn depends on the load.

Therefore, the standard theorem fails.

Students who blindly apply $R_L = R_{\text{int}}$ will either use the ambient resistance ( $1 \ \Omega$ ) or incorrectly solve for the steady-state resistance under the false condition $R_L = R_{\text{int}}$ (yielding $\approx 3.37 \ \Omega$ ).

The correct approach requires expressing the load power as a fully coupled function of the system's thermal state and maximizing it from first principles using calculus.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Maximum Power Transfer
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free