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Question #43329

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The Question

A very long ideal solenoid of radius $R$ and $n$ turns per unit length carries a high-frequency alternating current $I(t) = I_0 \cos(\omega t)$ . The interior of the solenoid is completely filled with a non-magnetic, non-conducting dielectric material of relative permittivity $\epsilon_r$ . The speed of light in vacuum is $c$ . Assuming $\omega R \ll c$ , which of the following expressions best represents the effective inductive reactance per unit length of the solenoid when accounting for the first-order correction due to Maxwell's displacement current?

Options

A.
$\mu_0 n^2 \pi R^2 \omega \left( 1 + \frac{\epsilon_r \omega^2 R^2}{8 c^2} \right)$
B.
$\mu_0 n^2 \pi R^2 \omega \left( 1 + \frac{\epsilon_r \omega^2 R^2}{4 c^2} \right)$
C.
$\mu_0 n^2 \pi R^2 \omega \left( 1 - \frac{\epsilon_r \omega^2 R^2}{8 c^2} \right)$
D.
$\mu_0 n^2 \pi R^2 \omega$

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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

Inductive Reactance Dependency|Dielectric Constant (K) Shift|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

Students typically memorize the static formula $L = \mu_0 n^2 A$ and assume a dielectric only affects capacitance, incorrectly choosing the uncorrected reactance.

This question tests the deep coupling between Faraday's Law and the Maxwell-Ampere Law.

A time-varying magnetic field induces an electric field, which in a dielectric produces a displacement current.

This displacement current generates its own magnetic field, modifying the total flux linkage and thus the effective inductance.

Failing to account for the spatial variation of this displacement current or incorrectly integrating it leads to missing the $1/8$ geometric factor or the $\epsilon_r$ dependence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Inductive Reactance Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free