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Question #43367

Difficulty: 8/10 | 🟢 Free

The Question

A DC voltage source of electromotive force $E = 16 \text{ V}$ and internal resistance $r = 1 \text{ \Omega}$ is connected to a load consisting of a variable linear resistor $R$ and a non-linear electronic component $X$ connected in parallel. The current $I_X$ through the non-linear component depends on the voltage $V$ across it according to the relation $I_X = \alpha V^2$ , where $\alpha = 1 \text{ A V}^{-2}$ . What is the maximum possible electrical power that can be dissipated in the variable resistor $R$ ?

Options

A.
$20 \text{ W}$
B.
$24 \text{ W}$
C.
$32 \text{ W}$
D.
$64 \text{ W}$

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

Maximum Power Transfer|Kirchhoff’s Junction Rule|Conservative vs Non-conservative

Why This Question Is Tricky

Common Mistakes

This question tests a student's fundamental understanding of power dissipation and circuit optimization, specifically targeting the misapplication of the Maximum Power Transfer Theorem (MPTT).

Students commonly fall into the trap of blindly applying the standard MPTT formula $P_{max} = E^2 / (4r) = 64 \text{ W}$ , or they incorrectly assume the linear resistor $R$ should simply be matched to the internal resistance $r$ ( $R = 1 \text{ \Omega}$ ).

However, the standard MPTT derivation assumes a purely linear, isolated load.

Because the load network contains a non-linear component ( $I_X \propto V^2$ ) in parallel, the effective resistance of the circuit is voltage-dependent, and the current shunted by the non-linear component cannot be ignored.

To solve this, the student must abandon the $E^2/4r$ shortcut, express the power delivered to $R$ explicitly as a function of the junction voltage using Kirchhoff's Laws, and perform direct calculus optimization.

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