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Step-by-Step Solution
Step 1: Identify the Relevant Physical Concepts
We are given monochromatic light of wavelength 667 nm and power 9 mW. Each photon of this light has energy given by the formula
$E_{\text{photon}} = \frac{hc}{\lambda}$,
where
$h$ is Planck’s constant and
$c$ is the speed of light in vacuum. The number of photons emitted per second is found by dividing the total power (energy per second) by the energy of one photon.
Step 2: Express Number of Photons per Second
If $P$ is the power of the laser beam, then total energy emitted per second is $P$ joules. The energy per photon is $\frac{hc}{\lambda}$. Therefore, the number of photons emitted per second $N$ is
$$
N = \frac{P}{E_{\text{photon}}} = \frac{P}{\frac{hc}{\lambda}} = \frac{P \,\lambda}{h\,c}.
$$
Step 3: Substitute Known Values
Use the following values (as typically cited in the problem statement):
$P = 9 \times 10^{-3}\,\text{W}$ (9 mW)
$\lambda = 667\,\text{nm} = 6.67 \times 10^{-7}\,\text{m}$
$h \approx 6.6 \times 10^{-34}\,\text{J\,s}$
$c = 3.0 \times 10^{8}\,\text{m s}^{-1}$
Substitute these values into the formula.
Step 4: Perform the Calculation
$$
N = \frac{(9 \times 10^{-3}) \times (6.67 \times 10^{-7})}{(6.6 \times 10^{-34}) \times (3 \times 10^8)}.
$$
Carry out the multiplication and division carefully:
Numerator = $9 \times 10^{-3} \times 6.67 \times 10^{-7} = 60.03 \times 10^{-10} = 6.003 \times 10^{-9}$.
Denominator = $6.6 \times 10^{-34} \times 3 \times 10^{8} = 19.8 \times 10^{-26} = 1.98 \times 10^{-25}$.
Thus,
$$
N = \frac{6.003 \times 10^{-9}}{1.98 \times 10^{-25}} \approx 3 \times 10^{16}.
$$
Step 5: State the Final Answer
The number of photons arriving per second on the target is
$$3 \times 10^{16}.$$
