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Question #43174

Difficulty: 8/10 | 🟢 Free

The Question

A high-frequency electromagnetic wave propagates in the $+z$ direction from vacuum ( $z < 0$ ) into a non-magnetic, lossless dielectric medium ( $z \ge 0$ ). The relative permittivity of the medium varies spatially as $\epsilon_r(z) = \left(1 + \frac{z}{L}\right)^2$ , where $L$ is a positive constant. The amplitude of the electric field of the wave at the boundary $z=0$ is $E_0$ . Assuming the frequency of the wave is sufficiently high such that the WKB approximation is valid (i.e., internal reflections are negligible), what is the amplitude of the magnetic field at a depth $z = 3L$ ? (Here $c$ is the speed of light in vacuum).

Options

A.
$\frac{2E_0}{c}$
B.
$\frac{4E_0}{c}$
C.
$\frac{E_0}{2c}$
D.
$\frac{E_0}{c}$

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

Magnitude Ratio Identity|Medium Dependency|Conservative Field Path Independence

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of electromagnetic wave propagation in non-homogeneous media, specifically the spatial dependence of the $E/B$ magnitude ratio.

A common trap is assuming that the vacuum identity $|E|/|B| = c$ holds universally, leading students to incorrectly calculate $|B| = E(z)/c = E_0/(2c)$ .

Another trap is assuming the electric field amplitude $E_0$ remains constant and only the velocity changes, yielding $|B| = n E_0/c = 4E_0/c$ .

The correct approach requires recognizing that the local wave impedance $\eta(z)$ dictates both the $E/B$ ratio and, via the conservation of the Poynting vector, the spatial attenuation of the electric field amplitude.

The magnetic field amplitude actually increases as $\sqrt{n(z)}$ while the electric field decreases as $1/\sqrt{n(z)}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Magnitude Ratio Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free