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Question #42859

Difficulty: 8/10 | 🟢 Free

The Question

An electromagnetic standing wave is established inside a one-dimensional cavity of length $L$ bounded by perfectly reflecting mirrors at $x = 0$ and $x = L$ . The cavity is filled with a non-magnetic dielectric medium whose refractive index varies spatially as $n(x) = n_0 \left(1 + \frac{x}{L}\right)$ , where $n_0$ is a positive dimensionless constant. The standing wave is excited in a resonant mode such that exactly two intermediate nodes exist between the mirrors (excluding the boundaries). What is the position $x_1$ of the first intermediate node from the mirror at $x = 0$ ?

Options

A.
$(\sqrt{2} - 1)L$
B.
$\frac{L}{3}$
C.
$(\sqrt{3} - 1)L$
D.
$\frac{L}{2}$

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

Standing Wave Nodes/Antinodes|Refractive Index Relativity|Phase and Path Difference

Why This Question Is Tricky

Common Mistakes

Students often memorize that nodes in a standing wave are separated by $\lambda/2$ , leading them to assume nodes are equidistant.

In a non-homogeneous medium, the wave speed $v(x) = c/n(x)$ and local wavelength $\lambda(x)$ vary spatially.

The trap is applying the uniform medium formula $x_m = m L / N$ , which incorrectly yields $L/3$ for the first node.

The correct approach requires using the Eikonal approximation where phase accumulates as an integral $\int k(x) dx$ , forcing the nodes to cluster where the refractive index is higher (slower wave speed).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Standing Wave Nodes/Antinodes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free