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Question #42834

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, thin rod of mass $M$ and length $L$ has a uniform positive charge $Q$ . The rod is placed along the $y$ -axis with its ends at $y = 0$ and $y = L$ , and is free to rotate about a frictionless pivot at the origin $(0,0)$ of a laboratory frame. A positive point charge $q$ is rigidly fixed at the coordinates $(-d, 0)$ in this frame. The laboratory frame accelerates with a constant acceleration $\vec{a} = a_0 \hat{i}$ relative to an inertial frame. If the rod is released from rest relative to the laboratory frame, what must be the magnitude of the acceleration $a_0$ such that the initial angular acceleration of the rod is zero? (Ignore gravity and assume $k = \frac{1}{4\pi\epsilon_0}$ )

Options

A.
$\frac{2 k q Q}{M L^2} \left( 1 - \frac{d}{\sqrt{d^2 + L^2}} \right)$
B.
$\frac{2 k q Q}{M L d}$
C.
$\frac{k q Q}{M d \sqrt{d^2 + L^2}}$
D.
$\frac{k q Q}{M L^2} \ln\left(1 + \frac{L^2}{d^2}\right)$

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

Continuous Charge Distribution|Infinite Wire vs. Finite Wire|Pseudo Force Directionality

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of torque integration for non-uniform force fields, specifically targeting the 'center of mass' fallacy.

Students often correctly calculate the net electric force but incorrectly assume it acts at the rod's center of mass (Option C).

Another common trap is blindly applying the infinite wire formula $E = 2k\lambda/d$ (Option B).

Furthermore, Option D traps students who ignore the vector cross product $\vec{r} \times d\vec{F}$ , erroneously integrating the scalar Coulomb force multiplied by distance, which yields a logarithmic function.

The correct approach requires integrating both the pseudo-force density and the electric force density to find the exact torques about the pivot.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Continuous Charge Distribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free