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Question #42829

Difficulty: 8/10 | 🟢 Free

The Question

A rigid electric dipole of dipole moment magnitude $p$ is moved slowly along the x-axis from the origin $(0,0)$ to the point $(L,0)$ . The region contains a static electric field given by $\vec{E} = (E_0 + \alpha x)\hat{i} - \alpha y \hat{j}$ , where $E_0$ and $\alpha$ are positive constants. During the motion, an external mechanism continuously rotates the dipole such that its angle $\theta$ with the positive x-axis changes linearly with its position $x$ , according to the relation $\theta(x) = \frac{\pi}{2} \left(1 - \frac{x}{L}\right)$ . What is the work done strictly by the translational electric force on the dipole during this entire process?

Options

A.
$\frac{2p\alpha L}{\pi}$
B.
$p(E_0 + \alpha L)$
C.
$p\alpha L$
D.
$p E_0 + \frac{2p\alpha L}{\pi}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dipole in Uniform Field|Variable Force Integration|Dipole in Non-Uniform Field

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between the total work done by an electric field and the work done specifically by the translational force.

Students commonly misapply the potential energy formula $U = -\vec{p}\cdot\vec{E}$ , assuming that the change in potential energy $-\Delta U$ gives the translational work.

However, $-\Delta U$ represents the total work done by the electric field, which is the sum of the work done by the translational force ( $\int \vec{F}\cdot d\vec{r}$ ) and the work done by the torque ( $\int \vec{\tau}\cdot d\vec{\theta}$ ).

By blindly using $W = \vec{p}_f\cdot\vec{E}_f - \vec{p}_i\cdot\vec{E}_i$ , students will incorrectly calculate $p(E_0 + \alpha L)$ and select the trap option B.

The correct approach requires integrating the spatial gradient of the field which is the actual source of the translational force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dipole in Uniform Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free