Before You Begin

Question #43239

Difficulty: 8/10 | 🟢 Free

The Question

A point source located at the origin emits continuous transverse waves. The waves propagate spherically outward through a non-ideal, isotropic medium. Due to the interaction of secondary wavelets with the medium, energy is continuously dissipated. The rate of total power loss with respect to the radial distance $r$ is given by $\frac{dP}{dr} = - 8\pi \alpha r I(r)$ where $I(r)$ is the local wave intensity and $\alpha$ is a dimensionless positive constant. If the amplitude of the wave at a distance $r_0$ is $A_0$ , what is the amplitude of the wave at a distance $2r_0$ ?

Options

A.
$A_0 2^{-(1-\alpha)}$
B.
$A_0 2^{-(1+\alpha)}$
C.
$A_0 2^{-2(1+\alpha)}$
D.
$A_0 2^{-1} e^{-3\alpha r_0^2}$

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

Huygens Principle Wavefronts|Energy Density in Transverse Waves|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental understanding of Huygens' principle and energy conservation in waves.

Students often blindly apply the inverse-square law for intensity ( $I \propto 1/r^2$ ) or the inverse law for amplitude ( $A \propto 1/r$ ) without accounting for the medium's continuous energy dissipation.

A common trap is failing to couple the geometric spreading (the $4\pi r^2$ area factor) with the energy loss, leading them to incorrectly integrate the intensity directly as if it were a 1D wave.

By requiring the integration of the total power $P(r)$ over the expanding wavefront, the problem forces students to synthesize geometric aberration with non-conservative damping.

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