Monochromatic light of frequency $6 \times 10^{14}$ Hz is produced by a laser. The power emitted is $2 \times 10^{-3}$ W. How many photons per second on average are emitted by the source.(Given $h = 6.63 \times 10^{-34} \, \text{Js}$)
- $9 \times 10^{18}$
- $6 \times 10^{15}$
- $5 \times 10^{15}$
- $7 \times 10^{16}$
The Correct Option is C
Approach Solution - 1
To find out how many photons are emitted per second by the laser, we need to calculate the energy of a single photon and then determine how many such photons correspond to the total power emitted.
The energy \(E\) of a single photon can be determined using Planck's equation:
\(E = h \times f\)
Where:
- \(h = 6.63 \times 10^{-34} \, \text{Js}\) is Planck's constant.
- \(f = 6 \times 10^{14} \, \text{Hz}\) is the frequency of the light.
Substituting the given values into the formula:
\(E = 6.63 \times 10^{-34} \times 6 \times 10^{14}\)
\(E = 3.978 \times 10^{-19} \, \text{J}\)
This is the energy of one photon. The total power \(P\) emitted by the laser is \(2 \times 10^{-3} \, \text{W}\) (which is joules per second).
We can find the number of photons emitted per second by dividing the total power by the energy of a single photon:
\(n = \frac{P}{E} = \frac{2 \times 10^{-3}}{3.978 \times 10^{-19}}\)
Calculating this, we find:
\(n \approx 5.03 \times 10^{15}\)
Therefore, the number of photons emitted per second is approximately \(5 \times 10^{15}\).
Thus, the correct answer is \(5 \times 10^{15}\), which matches with one of the given options.
Approach Solution -2
Given: - Frequency of light: \( \nu = 6 \times 10^{14} \, \text{Hz} \) - Power emitted by the source: \( P = 2 \times 10^{-3} \, \text{W} \) - Planck's constant: \( h = 6.63 \times 10^{-34} \, \text{Js} \)
Step 1: Calculating the Energy of One Photon
The energy \( E \) of a photon is given by:
\[ E = h\nu \]
Substituting the given values:
\[ E = 6.63 \times 10^{-34} \times 6 \times 10^{14} \, \text{J} \] \[ E = 3.978 \times 10^{-19} \, \text{J} \]
Rounding off:
\[ E \approx 4 \times 10^{-19} \, \text{J} \]
Step 2: Calculating the Number of Photons Emitted per Second
The number of photons emitted per second (\( n \)) is given by:
\[ n = \frac{P}{E} \]
Substituting the given values:
\[ n = \frac{2 \times 10^{-3}}{4 \times 10^{-19}} \] \[ n = \frac{2}{4} \times 10^{16} \] \[ n = 0.5 \times 10^{16} \] \[ n = 5 \times 10^{15} \]
Conclusion: The number of photons emitted per second by the source is \( 5 \times 10^{15} \).
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