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Question #43105

Difficulty: 8/10 | 🟢 Free

The Question

A solid right circular cone of base radius $R$ and height $H = \sqrt{3}R$ is made of a material of uniform density $\sigma$ . The cone is completely submerged in a transparent liquid of density $\rho$ (where $\rho < \sigma$ ). The curved surface of the cone is perfectly reflecting. The cone is oriented with its axis vertical and its apex pointing downwards. A uniform, parallel laser beam of intensity $I$ is directed vertically upwards from below, illuminating the entire curved surface of the cone. If the cone remains in stationary equilibrium, what is the required intensity $I$ of the laser beam? (Assume the momentum flux of the laser beam in the liquid is given by $I/c$ , where $c$ is the speed of light).

Options

A.
$\frac{c R (\sigma - \rho) g}{\sqrt{3}}$
B.
$\frac{2c R (\sigma - \rho) g}{\sqrt{3}}$
C.
$\frac{c R (\sigma - \rho) g}{2\sqrt{3}}$
D.
$\frac{2c R (\sigma - \rho) g}{3\sqrt{3}}$

Check Your Answer

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

Radiation Pressure|Intensity and Photon Flux|Archimedes' Buoyancy Dynamics|Integration

Why This Question Is Tricky

Common Mistakes

The core trap in this problem lies in the 'Vector Integral Overlap' for radiation pressure on a curved surface.

Students often erroneously assume the radiation force on a perfectly reflecting object is simply $2I/c \times (\text{Projected Area})$ , which is only true for a flat disk normal to the beam.

For a cone, the local normal angle dictates both the incident power per unit area and the direction of the reflected momentum.

By failing to account for the $\sin^2\alpha$ geometric dependence (or by misapplying Archimedes' principle to treat radiation pressure as a volumetric scalar field), students arrive at incorrect intensity values.

The elegant bypass is using global momentum conservation of the photon stream, recognizing the reflected beam diverges at an angle $2\alpha$ from the incident axis.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Radiation Pressure
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free