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Question #43568

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wavefront traveling in a vacuum grazes the apex of a perfectly absorbing, opaque obstacle. The cross-section of the obstacle in the $xy$ -plane is given by the curve $y = \frac{2}{3\sqrt{L_0}} x^{3/2}$ for $x \ge 0$ , where $L_0$ is a positive characteristic length. According to the Geometrical Theory of Diffraction, the grazing ray excites a creeping wave at the origin $(0,0)$ at time $t=0$ . This creeping wave travels along the surface of the obstacle at the speed of light $c$ , continuously shedding diffracted rays tangentially into the surrounding vacuum. At time $t = \frac{20 L_0}{3 c}$ , what is the radius of curvature of the diffracted wavefront (in the $xy$ -plane) at the position reached by the specific ray that was shed from the point $x = 3L_0$ on the obstacle?

Options

A.
$2 L_0$
B.
$(2 + 16\sqrt{3}) L_0$
C.
$16\sqrt{3} L_0$
D.
$\frac{32\sqrt{3}}{2 + 16\sqrt{3}} L_0$

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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

Huygens Principle Wavefronts|Flux and Surface Geometry|Potential Gradient Direction

Why This Question Is Tricky

Common Mistakes

Students often misapply geometrical optics (like the mirror or lens formulas) to diffraction problems, incorrectly assuming the radius of curvature of the diffracted wavefront depends on the local radius of curvature of the obstacle (which here is $R_{obs} = 16\sqrt{3}L_0$ ).

They fail to recognize that a creeping wave creates an involute wavefront.

Because the secondary Huygens sources are excited sequentially with a phase delay exactly matching the arc length, the resulting envelope's radius of curvature is purely the unwound path length $s = ct - l$ , completely independent of the obstacle's local curvature.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Huygens Principle Wavefronts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free