The output power of a laser pointer is about \(1 \mathrm{mW}\) (a) What are the energy and momentum of one laser photon if the laser wavelength is \(670 \mathrm{nm} ?\) (b) How many photons per second are emitted by the laser? (c) What is the average force on the laser due to the momentum carried away by these photons?

To find the energy of a single photon, we will use Planck's equation: \(E = hf\), where \(E\) is the energy, \(h\) is Planck's constant (\(6.63 \times 10^{-34} \mathrm{Js}\)), and \(f\) is the frequency of the light. First, we need to find the frequency using the given wavelength (\(\lambda = 670 \mathrm{nm}\)) and the speed of light (\(c = 3.0 \times 10^8 \mathrm{m/s}\)). The frequency is given by \(f = \frac{c}{\lambda}\).

To convert the wavelength to frequency, we will first convert the wavelength from nanometers to meters: \(\lambda = 670 \times 10^{-9} \mathrm{m}\). Next, use the speed of light to find the frequency: \(f = \frac{3.0 \times 10^8 \mathrm{m/s}}{670 \times 10^{-9} \mathrm{m}} = 4.48 \times 10^{14} \mathrm{Hz}\).

Now that we have the frequency, we can find the energy of a single photon using Planck's equation: \(E = hf = (6.63 \times 10^{-34} \mathrm{Js})(4.48 \times 10^{14} \mathrm{Hz}) = 2.97 \times 10^{-19} \mathrm{J}\).

The momentum of a photon is given by \(p = \frac{E}{c}\), where \(p\) is momentum, \(E\) is the calculated energy, and \(c\) is the speed of light. Therefore, \(p = \frac{2.97 \times 10^{-19} \mathrm{J}}{3.0 \times 10^8 \mathrm{m/s}} = 9.9\times 10^{-28} \mathrm{kg \cdot m/s}\).

We are given the output power of the laser pointer as \(1 \mathrm{mW} = 1 \times 10^{-3} \mathrm{W}\). We previously calculated the energy of a single photon. The power is the energy per second, so we can find the number of photons emitted per second by dividing the given power by the energy of a single photon: \(n = \frac{1 \times 10^{-3} \mathrm{W}}{2.97 \times 10^{-19} \mathrm{J}} = 3.36 \times 10^{15} \mathrm{photons/s}\).

Since we have the momentum of a single photon and the number of photons emitted per second, we can find the total momentum being carried away by the photons per second by multiplying these two quantities: \(\Delta p = 3.36 \times 10^{15} \mathrm{photons/s} \cdot 9.91 \times 10^{-28} \mathrm{kg \cdot m/s} = 3.33 \times 10^{-12} \mathrm{kg \cdot m/s^2}\). The average force on the laser pointer due to the momentum carried away by the photons is equal to the rate of change of momentum: \(F = \frac{\Delta p}{\Delta t} = 3.33 \times 10^{-12} \mathrm{N}\). The results are: (a) The energy of one laser photon is \(2.97 \times 10^{-19} \mathrm{J}\), and its momentum is \(9.91 \times 10^{-28} \mathrm{kg \cdot m/s}\). (b) The number of photons emitted per second is \(3.36 \times 10^{15} \mathrm{photons/s}\). (c) The average force on the laser pointer due to the momentum carried away by the photons is \(3.33 \times 10^{-12} \mathrm{N}\).