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Question #43575

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The Question

A beam of electrons (mass $m$ , charge $-e$ , initial velocity $v_0 \hat{i}$ ) is incident on a double-slit apparatus. The slits are separated by a distance $d$ along the y-axis (located at $y = \pm d/2, z=0$ ). A screen is placed at a distance $D$ from the slits at $x=D$ . A uniform longitudinal magnetic field $\vec{B} = B \hat{i}$ is applied in the region between the slits and the screen. Assuming $d \ll D$ and that the transverse velocity of the electrons is small compared to $v_0$ , what is the new fringe width $\beta$ of the interference pattern observed on the screen along the y-axis at $z=0$ ?

Let $\beta_0 = \frac{h D}{m v_0 d}$ be the standard fringe width in the absence of the magnetic field, and $\theta = \frac{e B D}{m v_0}$ be the cyclotron phase angle completed by the electrons during their transit.

Options

A.
$\beta = \beta_0 \frac{\theta}{2} \cot\left(\frac{\theta}{2}\right)$
B.
$\beta = \beta_0 \frac{2}{\theta} \tan\left(\frac{\theta}{2}\right)$
C.
$\beta = \beta_0 \cos\left(\frac{\theta}{2}\right)$
D.
$\beta = \beta_0 \left(1 + \frac{\theta^2}{4}\right)$

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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

Fringe Width Invariance|Trajectory of Charged Particle|Relative Velocity in 2D|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the profound misconception that the interference fringe width $\beta = \lambda D / d$ is a purely geometric constant invariant under external fields.

Students typically apply the standard straight-line path difference $\Delta x = y d / D$ without accounting for the helical trajectory induced by the longitudinal magnetic field.

The trap lies in ignoring that the Lorentz force couples the transverse velocity to the path length and introduces a magnetic vector potential phase shift.

By evaluating the classical action (or exact path length and enclosed flux), the effective transverse path difference becomes a non-linear function of the cyclotron angle $\theta$ , fundamentally altering the fringe width.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Fringe Width Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free