Before You Begin

Question #43759

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of uniform charge $Q$ (Coulombs), initial radius $R$ (meters), and bare mechanical mass $m$ (kg) is moving with a constant non-relativistic velocity $v_0$ (m/s) along the x-axis in a field-free vacuum. An internal mechanism causes the sphere to slowly expand radially to a new radius $2R$ while maintaining a uniform charge distribution. Assume no external forces act on the system and radiation losses are negligible. Let $c$ (m/s) be the speed of light in vacuum and $\epsilon_0$ (F/m) be the vacuum permittivity. If the total momentum of the system is given by $\vec{P} = m\vec{v} + \vec{p}_{em}$ , where $\vec{p}_{em} = \epsilon_0 \int (\vec{E} \times \vec{B}) dV$ is the electromagnetic field momentum (with $\vec{E}$ and $\vec{B}$ being the fields generated by the sphere), what is the final velocity $v_f$ of the sphere?

Options

A.
$v_0$
B.
$v_0 \left( \frac{m + \frac{3Q^2}{20\pi\epsilon_0 R c^2}}{m + \frac{3Q^2}{40\pi\epsilon_0 R c^2}} \right)$
C.
$v_0 \left( \frac{m + \frac{Q^2}{5\pi\epsilon_0 R c^2}}{m + \frac{Q^2}{10\pi\epsilon_0 R c^2}} \right)$
D.
$v_0 \left( \frac{m + \frac{Q^2}{6\pi\epsilon_0 R c^2}}{m + \frac{Q^2}{12\pi\epsilon_0 R c^2}} \right)$

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

Instantaneous Nature|Energy Density Locality|Non-conducting Charged Sphere

Why This Question Is Tricky

Common Mistakes

This question tests the concept of electromagnetic field momentum and its role in the conservation of total momentum.

A common trap is the 'Self-Energy Paradox' where students either completely ignore the momentum carried by the electromagnetic field (Option A) or incorrectly assume the electromagnetic mass is simply $U_e/c^2$ (Option B).

By explicitly requiring the evaluation of $\vec{p}_{em} = \epsilon_0 \int (\vec{E} \times \vec{B}) dV$ , the problem forces students to derive the subtle $4/3$ factor ( $\vec{p}_{em} = \frac{4}{3c^2}U_e \vec{v}$ ) inherent to the classical electron model.

Furthermore, Option D traps students who carelessly use the self-energy of a spherical shell instead of a solid sphere.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Instantaneous Nature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free