Before You Begin

Question #43106

Difficulty: 8/10 | 🟢 Free

The Question

Two identical dust particles, A and B, each of rest mass $m_0$ , are initially at rest in free space. Particle A is located at the origin ( $x=0$ ) and particle B is located at $x=L$ . At $t = 0$ , particle A begins to emit a perfectly collimated laser beam of constant power $P$ directed along the positive x-axis towards particle B. Particle B perfectly absorbs all the radiation incident on it. Assuming $P t \ll m_0 c^2$ (so that the rest masses of the particles can be considered constant), what is the velocity vector $\vec{v}_{cm}$ of the center of mass of the two-particle system (A and B) for time $t > L/c$ ? (Here $c$ is the speed of light in vacuum).

Options

A.
$-\frac{P L}{2 m_0 c^2} \hat{i}$
B.
$\vec{0}$
C.
$\frac{P L}{2 m_0 c^2} \hat{i}$
D.
$-\frac{P t}{2 m_0 c} \hat{i}$

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

Radiation Pressure|Center of Mass (COM) Frame|Photon Neutrality|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the conservation of momentum to a system involving electromagnetic radiation.

A common trap is to treat the radiation pressure purely as an internal action-reaction pair that instantaneously cancels out, leading to the incorrect conclusion that the center of mass of the two particles remains at rest ( $\vec{v}_{cm} = \vec{0}$ ).

However, because the photons travel at a finite speed $c$ , there is a time delay $L/c$ during which particle A recoils before particle B begins to absorb.

This creates a steady-state photon field that carries a constant momentum $PL/c^2$ .

To conserve the total momentum of the isolated system, the particles must acquire an equal and opposite momentum, resulting in a constant drift of their center of mass.

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