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Question #43426

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The Question

A circular non-conducting ring of radius $R$ carries a uniformly distributed total positive charge $Q$ in its rest frame $S'$ . The ring rotates with a constant angular velocity $\omega$ about its central axis (the $z$ -axis). The entire ring is then translated with a constant velocity $\vec{v} = v_0 \hat{i}$ ( $v_0 \ll c$ ) relative to the laboratory frame $S$ . Due to the relativistic transformation of linear charge density, the charge density in the laboratory frame is given by $\lambda = \gamma \left(\lambda' + \frac{v_0}{c^2} J'_x\right)$ , where $\lambda'$ is the rest frame charge density and $J'_x$ is the $x$ -component of the current density in the rest frame. If a uniform external electric field $\vec{E} = E_0 \hat{i}$ exists in the laboratory frame, what is the effective electric dipole moment $\vec{p}$ of the ring and the resulting electrostatic torque $\vec{\tau}$ experienced by it in the laboratory frame? (Assume $\gamma \approx 1$ for $v_0 \ll c$ )

Options

A.
$\vec{p} = -\frac{v_0 Q \omega R^2}{2 c^2} \hat{j}$ , $\vec{\tau} = \frac{v_0 Q \omega R^2 E_0}{2 c^2} \hat{k}$
B.
$\vec{p} = \frac{v_0 Q \omega R^2}{2 c^2} \hat{j}$ , $\vec{\tau} = -\frac{v_0 Q \omega R^2 E_0}{2 c^2} \hat{k}$
C.
$\vec{p} = -\frac{v_0 Q \omega R^2}{c^2} \hat{j}$ , $\vec{\tau} = \frac{v_0 Q \omega R^2 E_0}{c^2} \hat{k}$
D.
$\vec{p} = 0$ , $\vec{\tau} = 0$

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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

Magnetic Moment of Orbiting Electron|Frame of Reference Dependency|Lorentz Force Vector Direction

Why This Question Is Tricky

Common Mistakes

This question tests the relativistic transformation of charge density and its macroscopic consequences on a moving magnetic dipole.

A common Galilean misconception is that a uniformly charged rotating ring (a pure magnetic dipole) remains purely magnetic when translated, leading students to incorrectly choose zero torque.

However, the relativity of simultaneity dictates that the moving current distribution acquires an asymmetric charge density, manifesting as an effective electric dipole moment $\vec{p} = (\vec{v} \times \vec{\mu}_0)/c^2$ .

By providing the transformation rule, the problem evaluates the student's ability to integrate a non-uniform charge distribution and apply vector mechanics to find the resulting electrostatic torque, a concept central to resolving advanced electrodynamic paradoxes.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Moment of Orbiting Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free