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Question #43100

Difficulty: 8/10 | 🟢 Free

The Question

Two coherent plane electromagnetic waves, Wave 1 and Wave 2, propagate in a vacuum. Wave 1 propagates along the positive $z$ -axis with its electric field given by $\vec{E}_1 = E_0 \cos(kz - \omega t) \hat{i}$ . Wave 2 propagates along the positive $y$ -axis with its electric field given by $\vec{E}_2 = E_0 \cos(ky - \omega t) \hat{i}$ . Let $\langle u \rangle$ represent the time-averaged total energy density of the superposed electromagnetic field at a given point in space. What is the ratio of the maximum time-averaged energy density to the minimum time-averaged energy density, $\frac{\langle u \rangle_{\text{max}}}{\langle u \rangle_{\text{min}}}$ , in the region of superposition?

Options

A.
$3$
B.
$\infty$
C.
$2$
D.
$1$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Energy Density Partitioning|Phase and Path Difference|Intensity Distribution in YDSE

Why This Question Is Tricky

Common Mistakes

This question targets the misconception that the total energy density (or intensity) of superposed EM waves can be calculated by treating intensity as a simple scalar that scales with the square of the total electric field ( $I \propto E^2$ ).

Students falling for this trap assume that at points of complete destructive interference for the electric field, the total energy density must be zero, leading to an infinite ratio (Option B).

However, energy density is a quadratic form partitioned into electric and magnetic components ( $u = \frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$ ).

Because the waves propagate in orthogonal directions, their magnetic fields are orthogonal and do not produce an interference cross-term.

Consequently, at electric nodes, the local energy density is purely magnetic and non-zero, demonstrating that E and B components must be analyzed independently for 3D superposition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Energy Density Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free