29.8 The particle-wave duality reviewed  (Page 2/4)

Integrated concepts

The problem set for this section involves concepts from this chapter and several others. Physics is most interesting when applied to general situations involving more than a narrow set of physical principles. For example, photons have momentum, hence the relevance of Linear Momentum and Collisions . The following topics are involved in some or all of the problems in this section:

• Dynamics: Newton’s Laws of Motion
• Work, Energy, and Energy Resources
• Linear Momentum and Collisions
• Heat and Heat Transfer Methods
• Electric Potential and Electric Field
• Electric Current, Resistance, and Ohm’s Law
• Wave Optics
• Special Relativity

[link] illustrates how these strategies are applied to an integrated-concept problem.

Recoil of a dust particle after absorbing a photon

The following topics are involved in this integrated concepts worked example:

A 550-nm photon (visible light) is absorbed by a $1\text{.}\text{00-μg}$ particle of dust in outer space. (a) Find the momentum of such a photon. (b) What is the recoil velocity of the particle of dust, assuming it is initially at rest?

Strategy Step 1

To solve an integrated-concept problem , such as those following this example, we must first identify the physical principles involved and identify the chapters in which they are found. Part (a) of this example asks for the momentum of a photon , a topic of the present chapter. Part (b) considers recoil following a collision , a topic of Linear Momentum and Collisions .

Strategy Step 2

The following solutions to each part of the example illustrate how specific problem-solving strategies are applied. These involve identifying knowns and unknowns, checking to see if the answer is reasonable, and so on.

Solution for (a)

The momentum of a photon is related to its wavelength by the equation:

Entering the known value for Planck’s constant $h$ and given the wavelength $\lambda$ , we obtain

Discussion for (a)

This momentum is small, as expected from discussions in the text and the fact that photons of visible light carry small amounts of energy and momentum compared with those carried by macroscopic objects.

Solution for (b)

Conservation of momentum in the absorption of this photon by a grain of dust can be analyzed using the equation:

The net external force is zero, since the dust is in outer space. Let 1 represent the photon and 2 the dust particle. Before the collision, the dust is at rest (relative to some observer); after the collision, there is no photon (it is absorbed). So conservation of momentum can be written

where $p_1$ is the photon momentum before the collision and ${p\prime }_2$ is the dust momentum after the collision. The mass and recoil velocity of the dust are $m$ and $v$ , respectively. Solving this for $v$ , the requested quantity, yields

where $p$ is the photon momentum found in part (a). Entering known values (noting that a microgram is ${\text{10}}^{-9}\text{kg}$ ) gives