Monochromatic light of wavelength `667 nm` is produced by a helium neon laser . The power emitted is `9 mW` . The number of photons arriving per second on the average at a target irradiated by this beam is

Options

A. `9 xx 10^(17)`
B. `3 xx 10^(16)`
C. `9 xx 10^(15)`
D. `3 xx 10^(19)`

Explanation

To find the number of photons arriving per second at a target irradiated by a beam of monochromatic light from a helium neon laser, we can follow these steps:

### Step 1: Understand the relationship between power, energy, and number of photons
The power \( P \) of the laser light is related to the energy of the photons and the number of photons arriving per second \( \frac{N}{t} \) by the equation:
\[ P = \frac{N}{t} \cdot E \]
where \( E \) is the energy of a single photon.

### Step 2: Calculate the energy of a single photon
The energy \( E \) of a single photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \, \text{J s}) \)
- \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \)
- \( \lambda \) is the wavelength of the light in meters.

Given that the wavelength \( \lambda = 667 \, \text{nm} = 667 \times 10^{-9} \, \text{m} \), we can substitute these values into the equation.

### Step 3: Substitute the values into the energy equation
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \cdot (3 \times 10^8 \, \text{m/s})}{667 \times 10^{-9} \, \text{m}} \]

### Step 4: Calculate the energy of a single photon
Calculating this gives:
\[ E = \frac{(6.626 \times 10^{-34}) \cdot (3 \times 10^8)}{667 \times 10^{-9}} \approx 2.98 \times 10^{-19} \, \text{J} \]

### Step 5: Calculate the number of photons per second
Now, we can rearrange the power equation to find \( \frac{N}{t} \):
\[ \frac{N}{t} = \frac{P}{E} \]
Given that the power \( P = 9 \, \text{mW} = 9 \times 10^{-3} \, \text{W} \), we substitute this into the equation:
\[ \frac{N}{t} = \frac{9 \times 10^{-3}}{2.98 \times 10^{-19}} \]

### Step 6: Perform the calculation
Calculating this gives:
\[ \frac{N}{t} \approx 3.02 \times 10^{16} \, \text{photons/second} \]

### Final Answer
The number of photons arriving per second on the average at the target is approximately \( 3.02 \times 10^{16} \, \text{photons/second} \).

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