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Step-by-Step Solution
Step 1: Identify the Given Quantities
• Power of the laser, $P = 2 \,\text{mW} = 2 \times 10^{-3}\,\text{J/s}$
• Wavelength of emitted light, $\lambda = 500 \,\text{nm} = 500 \times 10^{-9}\,\text{m}$
• Planck's constant, $h = 6.6 \times 10^{-34}\,\text{J·s}$
• Speed of light, $c = 3.0 \times 10^{8}\,\text{m/s}$
We want the number of photons emitted per second, $\frac{dn}{dt}$.
Step 2: Write the Energy of a Single Photon
The energy $E$ of a single photon is given by:
$$
E = \frac{hc}{\lambda}
$$
Step 3: Relate Power to Number of Photons Emitted Per Second
Power $P$ is the total energy emitted per unit time. If $\frac{dn}{dt}$ is the number of photons emitted per second, then
$$
P = E \times \frac{dn}{dt} = \left(\frac{hc}{\lambda}\right)\frac{dn}{dt}.
$$
Rearranging to find $\frac{dn}{dt}$,
$$
\frac{dn}{dt} = \frac{P \,\lambda}{h\,c}.
$$
Step 4: Substitute the Numerical Values
Substitute $P = 2 \times 10^{-3}\,\text{J/s}$, $\lambda = 500 \times 10^{-9}\,\text{m}$, $h = 6.6 \times 10^{-34}\,\text{J·s}$, and $c = 3 \times 10^{8}\,\text{m/s}$:
$$
\frac{dn}{dt}
= \frac{\left(2 \times 10^{-3}\right)\left(500 \times 10^{-9}\right)}{\left(6.6 \times 10^{-34}\right)\left(3 \times 10^{8}\right)}.
$$
Step 5: Simplify the Expression
First compute the product in the numerator:
$$
2 \times 10^{-3} \times 500 \times 10^{-9} = 2 \times 500 \times 10^{-3-9}
= 1000 \times 10^{-12} = 10^{-9}.
$$
But proceed carefully step by step:
• $2 \times 500 = 1000$
• $10^{-3} \times 10^{-9} = 10^{-12}$
So the numerator becomes $1000 \times 10^{-12} = 10^{-9}\,\text{J·m}$.
Now the denominator is
$$
6.6 \times 10^{-34} \times 3 \times 10^{8}
= 19.8 \times 10^{-26}
= 1.98 \times 10^{-25}.
$$
Hence,
$$
\frac{dn}{dt} = \frac{10^{-9}}{1.98 \times 10^{-25}}.
$$
This simplifies to:
$$
\frac{dn}{dt} = \frac{1}{1.98} \times 10^{16} \approx 0.505 \times 10^{16}.
$$
Or approximately:
$$
\frac{dn}{dt} \approx 5.05 \times 10^{15} \approx 5 \times 10^{15}.
$$
Final Answer
The number of photons emitted per second is approximately
$$
5 \times 10^{15}.
$$
