Before You Begin

Question #43123

Difficulty: 8/10 | 🟢 Free

The Question

A uniform laser beam of intensity $I$ propagates in vacuum along the $+\hat{k}$ direction. It is incident on a transparent dielectric cone of base area $A$ and refractive index $n$ ( $n > 1$ ). The axis of the cone is aligned with the $z$ -axis, and its vertex points towards the laser source. The semi-vertical angle of the cone is $\alpha$ . Assume the surface of the cone has an ideal anti-reflection coating (100% transmission) and that the momentum of a photon in the dielectric medium is given by $p = nE/c$ , where $E$ is the photon energy. What is the net force vector $\vec{F}$ exerted by the laser beam on the cone?

Options

A.
$\frac{IA}{c} \sin\alpha \left( \sin\alpha - \sqrt{n^2 - \cos^2\alpha} \right) \hat{k}$
B.
$\frac{IA}{c} \cos\alpha \left( \cos\alpha - \sqrt{n^2 - \sin^2\alpha} \right) \hat{k}$
C.
$\frac{IA}{c} (1 - n) \hat{k}$
D.
$\frac{IA}{c} \sin\alpha \left( \sin\alpha + \sqrt{n^2 - \cos^2\alpha} \right) \hat{k}$

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

Snell’s Law Vector Form|Radiation Pressure|Vector Dot-Product

Why This Question Is Tricky

Common Mistakes

This question tests the vector nature of radiation pressure and the application of Minkowski momentum ( $p = nE/c$ ) in dielectrics.

Students commonly fall into the trap of treating momentum transfer as a scalar difference ( $p_i - p_t$ ) or incorrectly assuming the transmitted light continues along the $z$ -axis without accounting for the angular deviation.

By requiring the dot product of the transmitted wave vector with the incident direction, the problem forces the use of the vector form of Snell's Law.

Furthermore, it highlights a counter-intuitive physical reality: because the forward momentum of the light field increases upon entering the denser medium, the dielectric experiences a negative force (a pull towards the source), which is mathematically captured by the negative sign of the correct expression.

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