Before You Begin

Question #43156

Difficulty: 8/10 | 🟢 Free

The Question

An AC Wheatstone bridge ABCD is constructed with the following components in its four arms: Arm AB contains a resistor $R_1 = 10\ \Omega$ in series with an inductor $L = 20\ \text{mH}$ . Arm BC contains a pure resistor $R_2 = 50\ \Omega$ . Arm AD contains a pure resistor $R_3 = 40\ \Omega$ . Arm DC contains a resistor $R_4$ in series with a capacitor $C = 50\ \mu\text{F}$ . An AC voltage source $V = V_0 \sin(\omega t)$ is connected across terminals A and C. An ideal AC null detector is connected across terminals B and D. If the detector registers zero current, what are the required values of the resistance $R_4$ and the angular frequency $\omega$ ?

Options

A.
$R_4 = 200\ \Omega$ , $\omega = 1000\ \text{rad/s}$
B.
$R_4 = 160\ \Omega$ , $\omega = 1000\ \text{rad/s}$
C.
$R_4 = 160\ \Omega$ , $\omega = 250\ \text{rad/s}$
D.
$R_4 = 200\ \Omega$ , $\omega = 250\ \text{rad/s}$

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

Frequency Sensitivity of Z|Wheatstone Bridge Sensitivity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from DC Wheatstone bridge logic to AC impedance balance.

A common trap is applying the purely resistive DC balance condition ( $R_1/R_2 = R_3/R_4$ ) to find $R_4 = 200\ \Omega$ , while independently assuming the reactive components must cancel out via standard LC resonance ( $\omega = 1/\sqrt{LC} = 1000\ \text{rad/s}$ ).

However, in an AC bridge, the balance condition $Z_1 Z_4 = Z_2 Z_3$ requires both magnitude and phase to match simultaneously.

This cross-multiplication of complex impedances couples the real and imaginary parts, making the required resistance $R_4$ dependent on the reactive components ( $L$ and $C$ ), and the balance frequency $\omega$ dependent on the resistive components.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Frequency Sensitivity of Z
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free