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Question #43619

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ moves in a closed elliptical orbit around a fixed massive body of mass $M$ ( $M \gg m$ ). The semi-major axis of the orbit is $a$ and its eccentricity is $e$ . Let $\lambda$ be the instantaneous de Broglie wavelength of the particle. Which of the following expressions represents the integral $\oint \frac{1}{\lambda} ds$ evaluated over one complete orbit? (Here $ds$ is the infinitesimal arc length along the orbit, $G$ is the universal gravitational constant, and $h$ is Planck's constant).

Options

A.
$\frac{2\pi m}{h} \sqrt{GMa}$
B.
$\frac{2\pi m}{h} \sqrt{GMa(1-e^2)}$
C.
$\frac{2\pi m}{h} \sqrt{\frac{GMa}{1-e^2}}$
D.
$\frac{2\pi m}{h} \sqrt{GMa} (1-e^2)$

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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

De Broglie Wavelength Scaling|Orbital Velocity Energy Balance|Gravitation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link quantum mechanical wavelength scaling with classical orbital mechanics, specifically targeting the misconception that orbital velocity can be approximated by its transverse component.

A common trap is to incorrectly assume $v \approx v_\theta = L/(mr)$ , ignoring the radial velocity component $\dot{r}$ .

If a student makes this error, the integral $\int v^2 dt$ simplifies to $\int (L/mr)^2 dt = \int (L/mr) d\theta = 2\pi L/m$ , leading directly to the incorrect Option B, which contains the eccentricity term $(1-e^2)^{1/2}$ .

The correct approach requires transforming the spatial line integral into a time integral using $ds = v dt$ , and rigorously applying the vis-viva equation alongside Kepler's Second Law to evaluate the time-averaged inverse distance.

Quick Facts About This Question

Question Information

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