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Question #43479

Difficulty: 8/10 | 🟢 Free

The Question

A non-linear circuit element $X$ has a voltage-current characteristic given by $V_X = \alpha I^2$ , where $\alpha = 2 \ \text{V}\cdot\text{A}^{-2}$ and $I$ is the current flowing through it ( $I > 0$ ). The element $X$ is connected in series with a resistor of resistance $R = 2 \ \Omega$ , an ideal DC voltage source $V_{DC} = 12 \ \text{V}$ , and an ideal AC voltage source $v_{ac}(t) = v_0 \sin(\omega t)$ , where $v_0 = 60 \ \text{mV}$ . Assuming $v_0$ is sufficiently small so that the small-signal linear approximation is valid, what is the amplitude of the AC component of the current flowing through the circuit?

Options

A.
$6 \ \text{mA}$
B.
$10 \ \text{mA}$
C.
$7.5 \ \text{mA}$
D.
$15 \ \text{mA}$

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

Average vs. Instantaneous Calculus|Dynamic Resistance|Graphical Slope Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between static (average) resistance and dynamic (instantaneous) resistance in non-linear components.

A common trap is to calculate the static resistance at the operating point ( $R_{static} = V_X / I_{DC} = 8\ \text{V} / 2\ \text{A} = 4\ \Omega$ ) and use it for the AC signal analysis, which incorrectly yields an AC current amplitude of $10\ \text{mA}$ .

Students must recognize that for small-signal AC variations superimposed on a DC bias, the relevant opposition to current change is the local derivative of the voltage-current curve ( $r_d = dV/dI$ ), not the macroscopic ratio ( $V/I$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Average vs. Instantaneous Calculus
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free