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Question #42635

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q$ is initially at rest at the origin $(0,0)$ of a Cartesian coordinate system at time $t=0$ . It is placed in a region where a quasi-static, spatially uniform but time-varying electric field is present, given by: $\vec{E}(t) = E_0 \cos(\omega t) \hat{i} + E_0 \cos(2\omega t) \hat{j}$ where $E_0$ and $\omega$ are positive constants. Which of the following represents the equation of the trajectory of the particle in the $xy$ -plane?

Options

A.
$y = x - \frac{m\omega^2}{2qE_0} x^2$
B.
$y = \frac{m\omega^2}{2qE_0} x^2 - \frac{qE_0}{4m\omega^2}$
C.
$y = x + \frac{m\omega^2}{2qE_0} x^2$
D.
$y = \frac{m\omega^2}{2qE_0} x^2$

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

Common Mistakes

This question tests the vector superposition of non-parallel oscillating fields and the critical transition from 1D differential equations to 2D kinematics.

The primary trap lies in ignoring the constants of integration.

Students often incorrectly assume that an oscillating force proportional to $\cos(\omega t)$ directly implies a zero-centered SHM displacement $x \propto \cos(\omega t)$ , leading them to Option B.

However, because the particle starts from rest at the origin, the initial conditions dictate a drift component that shifts the center of oscillation.

This requires rigorous integration of the vector equation of motion, resulting in a Lissajous-like parabolic trajectory $y = x - cx^2$ rather than a simple symmetric parabola centered on the y-axis.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Superposition of SHMs
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free