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Question #42961

Difficulty: 8/10 | 🟢 Free

The Question

A thin toroidal coil of mean circumference $l = 1.0 \text{ m}$ and cross-sectional area $A = 1.0 \times 10^{-3} \text{ m}^2$ has $N = 1000$ turns of wire uniformly wound on a ferromagnetic core. The core material exhibits a non-linear magnetic behavior (ignoring hysteresis) where the magnetic field $B$ is related to the magnetic intensity $H$ by the equation $B = \frac{B_s H}{H_c + H}$ for $H > 0$ . Here, the saturation magnetic field is $B_s = 2.0 \text{ T}$ and the characteristic magnetic intensity is $H_c = 4000 \text{ A/m}$ . A time-varying current $I(t) = \alpha t$ is passed through the coil, where $\alpha = 2.0 \text{ A/s}$ and $t$ is the time in seconds. Calculate the magnitude of the induced EMF in the coil at $t = 2.0 \text{ s}$ .

Options

A.
$0.25 \text{ V}$
B.
$0.50 \text{ V}$
C.
$1.00 \text{ V}$
D.
$0.75 \text{ V}$

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

Soft vs Hard Magnetic Materials|Maxwell's Correction to Ampere's Law|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental understanding of Faraday's Law in non-linear magnetic media.

A common trap is for students to calculate the 'secant permeability' $\mu = \frac{B}{H}$ at $t=2.0\text{ s}$ (which yields $\frac{1}{4000}\text{ T m/A}$ ) and use it to find an effective inductance $L = \frac{\mu N^2 A}{l}$ , leading to an incorrect EMF of $\varepsilon = L \frac{dI}{dt} = 0.50\text{ V}$ .

Alternatively, they might use the initial permeability $\mu_i = \frac{B_s}{H_c}$ leading to $1.00\text{ V}$ .

The core physical principle is that induced EMF depends strictly on the time derivative of the flux, which in a non-linear medium requires the use of the 'differential permeability' $\mu_d = \frac{dB}{dH}$ .

This punishes the blind application of lumped-circuit formulas like $L = \frac{N\Phi}{I}$ and rewards a rigorous application of Maxwell-Faraday principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Soft vs Hard Magnetic Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free