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Question #43199

Difficulty: 8/10 | 🟢 Free

The Question

A point source emitting light of total power $P$ (in Watts) isotropically is placed at the origin $(0,0,0)$ . The source is embedded in a transparent medium of refractive index $\mu = \sqrt{3}$ . The boundary of this medium with vacuum is a paraboloidal surface given by the equation $z = H - \frac{x^2+y^2}{4H}$ (for $z \ge 0$ ), where $H$ is a positive constant in meters. The medium is bounded below by an opaque, perfectly absorbing plane at $z=0$ . Assuming that light rays incident on the paraboloidal boundary with an angle of incidence less than or equal to the critical angle are fully transmitted, and those exceeding it are totally internally reflected, what is the total light power that escapes into the vacuum?

Options

A.
$\frac{P}{3}$
B.
$\frac{P}{2} \left( 1 - \sqrt{\frac{2}{3}} \right)$
C.
$\frac{P}{4}$
D.
$\frac{P}{2} \left( 1 - \frac{1}{\sqrt{3}} \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Total Internal Reflection (TIR)|Flux and Surface Geometry|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of flux and solid angle in the context of geometric optics, specifically penalizing students who memorize the flat-interface 'circle of illuminance' formula ( $P_{esc} = P/2 (1-\cos\theta_c)$ ) and blindly apply it to curved boundaries.

By using a paraboloidal boundary, the local normal vector $\vec{n} = \nabla f$ changes continuously, forcing the student to calculate the local angle of incidence $\theta_i$ as a function of $z$ .

The correct approach recognizes the analogy between isotropic light emission and Gauss's Law, where the escaped power is strictly proportional to the solid angle subtended by the escaping region, not the surface area or a fixed critical cone.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Total Internal Reflection (TIR)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free