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Step 1: Identify the Given Data
• Wavelength of laser light, $ \lambda = 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m}$
• Power of the laser, $ P = 0.5 \, \text{kW} = 5 \times 10^{2} \, \text{W}$
• Duration (pulse width), $ t = 60 \, \text{ms} = 60 \times 10^{-3} \, \text{s}$
• Planck's constant, $ h = 6.62 \times 10^{-34} \, \text{J} \cdot \text{s}$
• Speed of light, $ c = 3 \times 10^{8} \, \text{m/s}$
Step 2: Calculate the Total Energy of the Pulse
The total energy $ E_{\text{pulse}} $ in the laser pulse is the product of its power and duration:
$ E_{\text{pulse}} = P \times t = (5 \times 10^{2}) \times (60 \times 10^{-3}) \, \text{J} $
$ E_{\text{pulse}} = 500 \times 0.06 \, \text{J} = 30 \, \text{J} $
Step 3: Calculate the Energy of One Photon
The energy of a single photon of wavelength $ \lambda $ is given by:
$ E_{\text{photon}} = \frac{hc}{\lambda} $
Substituting the values:
$ E_{\text{photon}}
= \frac{(6.62 \times 10^{-34}) \times (3 \times 10^{8})}{660 \times 10^{-9}} \, \text{J} $
Simplifying,
$ E_{\text{photon}}
= \frac{6.62 \times 3}{660} \times 10^{-34+8+9} \, \text{J}
= \frac{19.86}{660} \times 10^{-17} \, \text{J} $
This is approximately:
$ E_{\text{photon}} \approx 3 \times 10^{-19} \, \text{J} $
Step 4: Compute the Number of Photons in the Pulse
The total number of photons $ N $ is simply the total energy of the pulse divided by the energy per photon:
$ N = \frac{E_{\text{pulse}}}{E_{\text{photon}}}
= \frac{30}{3 \times 10^{-19}}
= 10^{20} $
Final Answer
The approximate number of photons in the pulse is $ 10^{20} $.
