Before You Begin

Question #43052

Difficulty: 8/10 | 🟢 Free

The Question

A beam of non-relativistic electrons, each of mass $m$ and initial kinetic energy $E$ , is incident on a double-slit apparatus with slit separation $d$ . A screen is placed at a distance $D$ from the slits ( $D \gg d$ ). In the region between the slits and the screen, a uniform electric field is applied perpendicular to the plane of the slits. This field accelerates the electrons such that their kinetic energy upon reaching the screen is $4E$ . If $\lambda_0$ is the initial de Broglie wavelength of the electrons before entering the slits, what is the fringe width $\beta$ of the interference pattern formed on the screen? (Assume the small-angle approximation holds and ignore gravitational effects).

Options

A.
$\frac{\lambda_0 D}{d}$
B.
$\frac{3}{4} \frac{\lambda_0 D}{d}$
C.
$\sqrt{\frac{2}{5}} \frac{\lambda_0 D}{d}$
D.
$\frac{2}{3} \frac{\lambda_0 D}{d}$

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

Optical Path Difference|De Broglie Wavelength Scaling|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the validity of the standard Young's Double Slit Experiment (YDSE) formula $\beta = \lambda D/d$ in a region with a spatially varying potential.

Students commonly fall into the trap of assuming they can use the initial wavelength (Option A), the arithmetic average of the initial and final wavelengths (Option B), or the wavelength corresponding to the average kinetic energy (Option C).

The rigorous physical approach requires calculating the phase accumulation via the action integral $\int \vec{p} \cdot d\vec{r}$ along the classical parabolic trajectory.

The effective momentum that dictates the interference pattern is derived from the spatial integral of the inverse momentum, which surprisingly yields the arithmetic mean of the initial and final momenta for a uniform field, leading to a non-intuitive fractional coefficient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Optical Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free