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Question #42849

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave is normally incident from vacuum ( $z < 0$ ) onto a non-uniform non-conducting medium ( $z > 0$ ). The incident electric field is given by $\vec{E}_i = E_0 \cos(kz - \omega t) \hat{y}$ . The medium in the region $z > 0$ has a uniform permittivity $\epsilon = \epsilon_0$ but a spatially varying magnetic permeability $\mu(x) = \mu_0 (1 + \alpha x)^2$ , where $\alpha > 0$ is a constant. Determine the reflected magnetic field $\vec{B}_r$ at the interface $z = 0$ , at the position $x = \frac{2}{\alpha}$ and time $t = 0$ .

Options

A.
$\frac{E_0}{2c} \hat{x}$
B.
$-\frac{E_0}{2c} \hat{x}$
C.
$\frac{E_0}{3c} \hat{x}$
D.
$-\frac{E_0}{c} \hat{x}$

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

Boundary Condition Reflection|Superposition of Magnetic Fields|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of electromagnetic boundary conditions in media with non-uniform magnetic permeability.

A common trap is for students to incorrectly assume the continuity of the tangential magnetic field $\vec{B}$ ( $B_{1t} = B_{2t}$ ) instead of the auxiliary field $\vec{H}$ ( $H_{1t} = H_{2t}$ ).

By treating $\mu$ as a scalar constant that cancels out, or ignoring its role in the local wave impedance $Z(x) = \sqrt{\mu(x)/\epsilon_0}$ , students will miscalculate the reflection coefficient.

Applying $B_{1t} = B_{2t}$ yields a reflected electric field of opposite sign ( $E_r = -E_0/2$ ), leading directly to the trap option $-\frac{E_0}{2c} \hat{x}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Boundary Condition Reflection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free