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Question #82966

Difficulty: 5/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $+q$ is initially at rest at the origin of a Cartesian coordinate system. At $t = 0$ , it is subjected to a time-varying electric field given by $\vec{E} = E_0 \cos(\omega t) \hat{i} + E_0 \cos(2\omega t) \hat{j}$ , where $E_0$ and $\omega$ are positive constants. The region is free of gravity and magnetic fields. What is the maximum kinetic energy attained by the particle during its subsequent motion?

Options

A.
$\frac{q^2 E_0^2}{2m\omega^2}$
B.
$\frac{27 q^2 E_0^2}{32m\omega^2}$
C.
$\frac{q^2 E_0^2}{m\omega^2}$
D.
$\frac{q^2 E_0^2}{2m\omega}$

Check Your Answer

Correct Answer

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

Superposition of SHMs|Electric Field Symmetry|Vector Cross-Products

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

The core misconception tested is the failure to treat the superposition of non-parallel oscillating fields as a 2D vector problem.

Students often fall into the trap of scalar addition, effectively treating the fields as collinear.

By doing so, they integrate the scalar sum of the fields ( $E_{eff} = E_0 \cos(\omega t) + E_0 \cos(2\omega t)$ ), leading to an incorrect maximization condition ( $\cos(\omega t) + \cos(2\omega t) = 0$ ) and an erroneous maximum kinetic energy of $\frac{27 q^2 E_0^2}{32 m \omega^2}$ (Option B).

The problem forces the transition from 1D differential equations to 2D vector kinematics, requiring independent integration of orthogonal components and subsequent vector magnitude maximization.

Option D serves as a dimensional analysis trap.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Superposition of SHMs
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free