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Question #43349

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The Question

A magnetic Wheatstone bridge is constructed to measure small fluctuations in magnetomotive force. Three of the arms are made of a linear magnetic material, each having a constant magnetic reluctance $\mathcal{R}_0$ . The fourth arm is a non-linear ferromagnetic core of length $l$ and uniform cross-sectional area $A$ . The material of the fourth arm follows the Fröhlich-Kennelly relation $B = \frac{\alpha H}{1 + \beta H}$ , where $B$ is the magnetic flux density, $H$ is the magnetic field intensity, and $\alpha$ and $\beta$ are positive constants. The fourth arm is biased by a steady DC magnetic field intensity $H_0$ . If the bridge is perfectly balanced for an infinitesimally small superimposed AC magnetomotive force, what is the required value of $\mathcal{R}_0$ ?

Options

A.
$\frac{l(1+\beta H_0)^2}{\alpha A}$
B.
$\frac{l(1+\beta H_0)}{\alpha A}$
C.
$\frac{l}{\alpha A}$
D.
$\frac{l(1+\beta H_0)^2}{\alpha \beta H_0 A}$

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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

Wheatstone Bridge Sensitivity|Magnetic Field Line Behavior in Matter|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of non-linear magnetic circuits and the critical distinction between static (secant) permeability and differential (tangent) permeability.

A common trap is to assume that the reluctance of the ferromagnetic arm is determined by its static permeability $\mu_s = B/H_0$ , which would lead to the incorrect balance condition $\mathcal{R}_0 = l(1+\beta H_0)/(\alpha A)$ (Option B).

However, for small-signal (AC) analysis superimposed on a DC bias, the system's response is governed by the local gradient of the B-H curve.

Students must recognize that the bridge sensitivity to small fluctuations depends strictly on the differential permeability $\mu_d = dB/dH$ , forcing a transition from linear algebraic reluctance to a calculus-based dynamic reluctance model.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Wheatstone Bridge Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free