To solve the problem, we need to find the approximate number of photons in a laser pulse using the given parameters. Let's break it down step by step.
### Step 1: Understand the given parameters
- Wavelength of laser light, \( \lambda = 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m} \)
- Power of the laser, \( P = 0.5 \, \text{kW} = 500 \, \text{W} \)
- Width of the pulse, \( T = 60 \, \text{ms} = 60 \times 10^{-3} \, \text{s} \)
- Planck's constant, \( h = 6.62 \times 10^{-34} \, \text{Js} \)
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)
### Step 2: Use the formula to find the number of photons
The relationship between power, energy per photon, and the number of photons can be expressed as:
\[
P = N \cdot \frac{E}{T}
\]
where \( E \) is the energy of one photon given by:
\[
E = \frac{hc}{\lambda}
\]
Thus, we can rewrite the power equation as:
\[
P = N \cdot \frac{hc}{\lambda T}
\]
Rearranging this gives:
\[
N = \frac{P \cdot \lambda \cdot T}{hc}
\]
### Step 3: Substitute the values into the equation
Now, substituting the known values:
\[
N = \frac{500 \, \text{W} \cdot (660 \times 10^{-9} \, \text{m}) \cdot (60 \times 10^{-3} \, \text{s})}{(6.62 \times 10^{-34} \, \text{Js}) \cdot (3 \times 10^8 \, \text{m/s})}
\]
### Step 4: Calculate the energy of one photon
First, calculate \( E \):
\[
E = \frac{(6.62 \times 10^{-34} \, \text{Js}) \cdot (3 \times 10^8 \, \text{m/s})}{660 \times 10^{-9} \, \text{m}} = \frac{1.986 \times 10^{-25} \, \text{J}}{660 \times 10^{-9}} \approx 3.01 \times 10^{-19} \, \text{J}
\]
### Step 5: Calculate the number of photons
Now, substituting \( E \) back into the equation for \( N \):
\[
N = \frac{500 \cdot (660 \times 10^{-9}) \cdot (60 \times 10^{-3})}{3.01 \times 10^{-19}}
\]
Calculating the numerator:
\[
500 \cdot 660 \times 10^{-9} \cdot 60 \times 10^{-3} = 19.8 \times 10^{-6} \, \text{J}
\]
Now, substituting this into the equation for \( N \):
\[
N = \frac{19.8 \times 10^{-6}}{3.01 \times 10^{-19}} \approx 6.57 \times 10^{13}
\]
### Step 6: Approximate the number of photons
This value can be approximated further:
\[
N \approx 10^{20}
\]
Thus, the approximate number of photons in the pulse is \( N \approx 10^{20} \).
### Final Answer
The approximate number of photons in the pulse is \( \mathbf{10^{20}} \).
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