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Question #42642

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave propagates in vacuum along the direction given by the unit vector $\hat{n} = \frac{\hat{i} + \hat{j} + \hat{k}}{\sqrt{3}}$ . A particle of mass $m$ and charge $q$ is released from rest at the origin. Neglecting radiation loss and gravity, and assuming the motion remains non-relativistic ( $v \ll c$ ), let $\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}$ be the instantaneous velocity of the particle at any time $t$ . Which of the following relations is always true?

Options

A.
$v_x^2 + v_y^2 + v_z^2 = \frac{2c}{\sqrt{3}} (v_x + v_y + v_z)$
B.
$v_x^2 + v_y^2 + v_z^2 = \frac{c}{\sqrt{3}} (v_x + v_y + v_z)$
C.
$v_x^2 + v_y^2 + v_z^2 = \frac{c}{2\sqrt{3}} (v_x + v_y + v_z)$
D.
$v_x^2 + v_y^2 + v_z^2 = \frac{\sqrt{3}c}{2} (v_x + v_y + v_z)$

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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

Particle Velocity vs. Wave Velocity|Magnitude Ratio Identity|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of the Lorentz force in an electromagnetic wave and the magnitude ratio identity $B = E/c$ .

The trap lies in attempting a brute-force 3D component resolution of the differential equations, which is mathematically overwhelming.

Students often incorrectly discard the magnetic force entirely in the non-relativistic limit, missing that it is the sole driver of longitudinal drift.

By using vector invariants—specifically the Work-Energy theorem and the rate of change of momentum along the wave vector—the exact relation emerges elegantly without needing to solve the complex coupled differential equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Particle Velocity vs. Wave Velocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free