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Question #43775

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The Question

A fixed, uniformly charged oblate spheroid with total charge $+Q$ is centered at the origin of a Cartesian coordinate system. Its axis of rotational symmetry lies along the $z$ -axis. A point charge $+q$ of mass $m$ is projected from a large distance ( $x \to -\infty$ ) with an initial velocity vector $\vec{v} = v_0 \hat{i}$ . The initial position of the particle is given by the coordinates $(-\infty, b, c)$ , where $b$ and $c$ are positive distances. Let $\vec{L}$ be the angular momentum vector of the particle about the origin. Which of the following correctly describes the conservation of angular momentum for this system during the particle's motion?

Options

A.
$\frac{d\vec{L}}{dt} = 0$ with $|\vec{L}| = m v_0 \sqrt{b^2 + c^2}$ , because the electrostatic field of an isolated charge distribution is a central force field.
B.
$\frac{dL_z}{dt} = 0$ with $L_z = -m v_0 b$ , while $\frac{d\vec{L}}{dt} \neq 0$ , because the electric field is azimuthally symmetric but non-central.
C.
$\frac{dL_y}{dt} = 0$ with $L_y = m v_0 c$ , while $\frac{d\vec{L}}{dt} \neq 0$ , because the particle's initial velocity is orthogonal to the $y$ -axis.
D.
$\frac{d\vec{L}}{dt} \neq 0$ and no individual Cartesian component of $\vec{L}$ is conserved, because the oblate spheroid produces a non-uniform, non-spherical potential.

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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

Impact Parameter Geometry|Gauss Law Applicability|Shell Theorem (Internal vs External)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between central forces and azimuthally symmetric non-central forces.

A common trap is assuming that any isolated charge distribution can be treated as a point charge (yielding a central force field) where total angular momentum $\vec{L}$ is conserved, leading students to incorrectly choose Option A.

Another trap is recognizing the non-spherical nature of the oblate spheroid but failing to apply Noether's theorem (or basic torque analysis) to realize that continuous rotational symmetry about the $z$ -axis guarantees the conservation of the $z$ -component of angular momentum ( $L_z$ ).

By forcing the evaluation of $\vec{r} \times \vec{F}$ , the problem highlights that while $\vec{E}$ is non-central ( $\vec{\tau} \neq 0$ ), the absence of an azimuthal field component ( $E_\phi = 0$ ) ensures $\tau_z = 0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Impact Parameter Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free