Before You Begin

Question #77724

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, non-conducting thin rod of length $L = 4a$ is placed along the positive x-axis with one end at the origin $(0,0,0)$ . The rod carries a non-uniform linear charge density $\lambda = c x^2$ , where $c$ is a positive constant with units of $\text{C}\cdot\text{m}^{-3}$ and $x$ is the distance from the origin. A point charge $q_0$ of mass $m$ is held stationary at the point $(0, 3a, 0)$ . A uniform magnetic field $\vec{B} = B_0 \hat{k}$ exists in the entire region. Determine the net initial force vector $\vec{F}$ experienced by the point charge $q_0$ immediately after it is released. Assume the permittivity of free space is $\epsilon_0$ .

Options

A.
$\frac{q_0 c a}{4\pi\epsilon_0} \left[ -\frac{4}{5} \hat{i} + \left(3\ln 3 - \frac{12}{5}\right) \hat{j} \right]$
B.
$\frac{q_0 c a}{4\pi\epsilon_0} \left[ \frac{4}{5} \hat{i} + \left(\ln 3 - \frac{4}{5}\right) \hat{j} \right]$
C.
$\frac{q_0 c a}{4\pi\epsilon_0} \left[ -\frac{4}{5} \hat{i} + 3\ln 3 \hat{j} \right]$
D.
$\frac{q_0 c a}{4\pi\epsilon_0} \left[ -\frac{8}{5} \hat{i} + \left(3\ln 3 - \frac{12}{5}\right) \hat{j} \right]$

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

Dimensional Independence vs. Scalar/Vector Nature|Coulomb's Law Vector Form|Integration

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Common Mistakes

This problem tests the dimensional independence of vector components during the integration of a non-uniform spatial charge density, a frequent stumbling block in JEE Advanced Electrostatics.

Students often fall into the trap of scalar summation, ignoring the spatial orientation of the differential force vector $d\vec{F}$ .

By introducing $\lambda = cx^2$ , the problem forces a rigorous vector projection where the x and y components require distinct integration techniques (algebraic substitution vs. trigonometric substitution).

A common mathematical trap is dropping the $-\cos\theta$ term during the y-component integration, leading directly to Option C.

Additionally, the presence of the magnetic field acts as a conceptual red herring, testing whether students correctly apply the Lorentz force law to recognize that a stationary charge experiences zero magnetic force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Independence vs. Scalar/Vector Nature
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free