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Question #43194

Difficulty: 8/10 | 🟢 Free

The Question

A beam of electrons, each of mass $m$ , is moving in the $x$ - $y$ plane in a region of zero potential energy ( $U=0$ ) with a constant velocity $\vec{v}_1 = v_0 \hat{i}$ . The beam encounters a straight potential step at the boundary defined by the line $\sqrt{3}x + y = 0$ . In the region $\sqrt{3}x + y > 0$ , the potential energy of the electrons is a constant $U_0 > 0$ . What is the minimum value of $U_0$ for which the electron beam cannot penetrate into the second region (i.e., undergoes total reflection)?

Options

A.
$\frac{1}{8} m v_0^2$
B.
$\frac{1}{4} m v_0^2$
C.
$\frac{3}{8} m v_0^2$
D.
$\frac{1}{2} m v_0^2$

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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

Snell’s Law Vector Form|Mechanical Energy Conservation|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply conservation laws at a discontinuous potential boundary, which acts as a refractive interface for matter waves.

The primary trap lies in misidentifying which component of momentum is conserved.

Students often incorrectly assume that the total kinetic energy $\frac{1}{2}mv_0^2$ must be overcome by the potential barrier (leading to Option D), ignoring that the tangential component of momentum cannot be destroyed by a normal force.

Another sophisticated trap is incorrectly applying an optical Snell's law analogy by assuming the 'refractive index' is proportional to the total energy $E$ rather than the momentum $p \propto \sqrt{E-U}$ , which leads exactly to Option B ( $\frac{1}{4}mv_0^2$ ).

The correct approach requires isolating the normal component of kinetic energy, as the barrier only exerts an impulsive force perpendicular to the interface.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Snell’s Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free