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Question #42268

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q$ is located at the origin of a coordinate system. At time $t=0$ , it is projected with an initial velocity $\vec{v} = v_0 \hat{i}$ . A time-dependent, spatially uniform electric field $\vec{E} = c t \hat{i}$ is present in the region, where $c$ is a positive constant. What is the work done by the electric field on the particle during the time interval from $t=0$ to $t=T$ ?

Options

A.
$\frac{q c v_0 T^2}{2} + \frac{q^2 c^2 T^4}{8m}$
B.
$\frac{q c v_0 T^2}{2}$
C.
$q c v_0 T^2 + \frac{q^2 c^2 T^4}{4m}$
D.
$\frac{q c v_0 T^2}{2} + \frac{q^2 c^2 T^4}{2m}$

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

Potential vs Field Relationship|Work-Energy Theorem Validity|Motion in a Straight Line

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle time-dependent fields where the concept of a static scalar electric potential is ill-defined.

A common trap is attempting to calculate work using $W = \int q \vec{E} \cdot d\vec{x}$ while incorrectly assuming the particle's trajectory is unperturbed ( $dx = v_0 dt$ ), leading to the partial answer.

By forcing the field to be time-varying ( $\partial \vec{E}/\partial t \neq 0$ ), students must abandon static potential gradients and directly apply the Work-Energy theorem, accounting for the field's continuous effect on the particle's velocity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Potential vs Field Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free