Integrated Concepts

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

Topics

Photons (quantum mechanics)

Linear Momentum

A 550-nm photon (visible light) is absorbed by a g\(1\text{.}\text{00-μg}\) 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 tutorials in which they are found. Part (a) of this example asks for the momentum of a photon, a topic of the present tutorial. 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:

\(p=\cfrac{h}{\lambda }.\)

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

\(\begin{array}{lll}p& =& \cfrac{\text{6.63}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot s}{\text{550}×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}\text{m}}\\ & =& 1\text{.}\text{21}×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}\text{.}\end{array}\)

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:

\({p}_{1}+{p}_{2}={p\prime }_{1}+{p\prime }_{2}({F}_{\text{net}}=0).\)

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

\({p}_{1}={p\prime }_{2}=\text{mv},\)

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

\(v=\cfrac{p}{m},\)

where \(p\) is the photon momentum found in part (a). Entering known values (noting that a microgram is \({\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{kg}\)) gives

\(\begin{array}{lll}v& =& \cfrac{1\text{.}\text{21}×{\text{10}}^{-\text{27}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot \text{m/s}}{1\text{.}\text{00}×{\text{10}}^{–9}\phantom{\rule{0.25em}{0ex}}\text{kg}}\\ & =& 1\text{.}\text{21}×{\text{10}}^{\text{–18}}\phantom{\rule{0.25em}{0ex}}\text{m/s.}\end{array}\)

Discussion

The recoil velocity of the particle of dust is extremely small. As we have noted, however, there are immense numbers of photons in sunlight and other macroscopic sources. In time, collisions and absorption of many photons could cause a significant recoil of the dust, as observed in comet tails.