A bulb of 40 W is producing a light of wavelength 620 nm with `80%` of efficiency , then the number of photons emitted by the bulb in 20 seconds are :
(`1eV = 1.6 xx 10^(-19) J , hc = 12400 eV`)

Options

A. `2 xx 10^(18)`
B. `10^(18)`
C. `10^(21)`
D. `2 xx 10^(21)`

Answer

To solve the problem, we need to calculate the number of photons emitted by a 40 W bulb producing light of wavelength 620 nm with 80% efficiency over a period of 20 seconds. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the total energy produced by the bulb in 20 seconds
The power of the bulb is given as 40 W. Power is defined as energy per unit time.

\[ \text{Energy} = \text{Power} \times \text{Time} \]

Given:
- Power = 40 W
- Time = 20 seconds

\[ \text{Energy} = 40 \, \text{W} \times 20 \, \text{s} = 800 \, \text{J} \]

### Step 2: Adjust for efficiency
The bulb operates at 80% efficiency, meaning only 80% of the total energy is converted into light.

\[ \text{Energy of light} = 0.80 \times 800 \, \text{J} = 640 \, \text{J} \]

### Step 3: Calculate the energy of one photon
To find the energy of one photon, we can use the formula:

\[ E = \frac{hc}{\lambda} \]

Where:
- \( h \) (Planck's constant) = \( 4.1357 \times 10^{-15} \, \text{eV s} \)
- \( c \) (speed of light) = \( 3 \times 10^8 \, \text{m/s} \)
- \( \lambda \) (wavelength) = 620 nm = \( 620 \times 10^{-9} \, \text{m} \)

Using the given value \( hc = 12400 \, \text{eV} \cdot \text{nm} \):

\[ E = \frac{12400 \, \text{eV} \cdot \text{nm}}{620 \, \text{nm}} = 20 \, \text{eV} \]

### Step 4: Convert the energy of one photon to joules
We need to convert the energy from electron volts to joules. Using the conversion \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):

\[ E = 20 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.2 \times 10^{-18} \, \text{J} \]

### Step 5: Calculate the number of photons emitted
The number of photons emitted can be calculated by dividing the total energy of light by the energy of one photon.

\[ \text{Number of photons} = \frac{\text{Energy of light}}{\text{Energy of one photon}} = \frac{640 \, \text{J}}{3.2 \times 10^{-18} \, \text{J}} \]

Calculating this gives:

\[ \text{Number of photons} = 2 \times 10^{21} \]

### Final Answer
The number of photons emitted by the bulb in 20 seconds is \( \mathbf{2 \times 10^{21}} \).

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