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Question #42875

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q > 0$ is initially at rest at the origin $(0,0,0)$ of an inertial reference frame. At $t=0$ , a plane electromagnetic wave is turned on. The electric field of the wave is given by $\vec{E} = E_0 \sin(kx - \omega t + \phi) \hat{j}$ and the magnetic field is $\vec{B} = B_0 \sin(kx - \omega t + \phi) \hat{k}$ , where $E_0 = c B_0$ and $c$ is the speed of light in vacuum. The phase constant $\phi$ is a given parameter ( $0 \le \phi < 2\pi$ ). Which of the following expressions represents the maximum longitudinal momentum ( $p_x$ ) acquired by the particle during its subsequent motion?

Options

A.
$\frac{q^2 E_0^2}{2 m c \omega^2} (1 + |\cos\phi|)^2$
B.
$\frac{q^2 E_0^2}{2 m c \omega^2} (1 + |\sin\phi|)^2$
C.
$\frac{q^2 E_0^2}{m c \omega^2} \cos^2\phi$
D.
$\frac{q^2 E_0^2}{4 m c \omega^2} (1 + \cos^2\phi)$

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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

Phase Constant (Epoch)|Lorentz Force Vector Direction|Velocity Vector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the ability to solve coupled non-linear equations of motion in an electromagnetic wave by identifying exact differentials and relativistic invariants.

Students often fall into the trap of assuming $v_x \approx 0$ to integrate the transverse force, or they incorrectly time-average the Lorentz force without accounting for the spatial phase coupling $\theta = kx - \omega t + \phi$ .

By recognizing that the phase derivative $d\theta/dt$ perfectly absorbs the velocity-dependent term $(1 - v_x/c)$ in the magnetic force, one can exactly integrate the transverse momentum $p_y$ .

Furthermore, the longitudinal momentum $p_x$ is inextricably linked to $p_y$ via the energy-momentum invariant $\mathcal{E} - p_x c = mc^2$ , demonstrating that the magnetic force dictates the exact longitudinal drift.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Phase Constant (Epoch)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free