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Question #43624

Difficulty: 8/10 | 🟢 Free

The Question

A parallel beam of light propagates in the $-z$ direction and strikes the curved surface of a perfectly reflecting hemisphere of radius $R$ . The flat base of the hemisphere lies in the $xy$ -plane with its center at the origin. The intensity of the incident beam is non-uniform and varies with the radial distance $ho = \sqrt{x^2 + y^2}$ from the $z$ -axis as $I(\rho) = I_0 \left(\frac{\rho}{R}\right)^2$ , where $I_0$ is a positive constant. If $c$ is the speed of light in vacuum, what is the magnitude of the net force exerted by the radiation on the hemisphere?

Options

A.
$\frac{\pi I_0 R^2}{3 c}$
B.
$\frac{8\pi I_0 R^2}{15 c}$
C.
$\frac{\pi I_0 R^2}{c}$
D.
$\frac{\pi I_0 R^2}{2 c}$

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

Intensity and Photon Flux|Radiation Pressure|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of radiation pressure on non-planar surfaces.

A common lethal trap is treating radiation pressure as a simple scalar field and integrating $P_{rad} dA_z$ , which incorrectly assumes the momentum transfer is identical to a flat surface (leading to Option C).

Another frequent error is failing to account for the three distinct trigonometric projections required: one for the projected area intercepting the beam ( $dA \cos\theta$ ), one for the normal component of the momentum change ( $\cos\theta$ ), and one for resolving the resulting normal force back along the axis of symmetry ( $\cos\theta$ ).

Missing the final resolution step leads to integrating $\cos^2\theta$ instead of $\cos^3\theta$ , trapping the student in Option B.

Option D traps students who incorrectly apply the formula for a perfectly absorbing surface.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Intensity and Photon Flux
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free