`10^(20)` photons of wavelength 660 nm are emitted per second from a lamp . What is the wattage of the lamp ?

Options

A. 30 W
B. 60 W
C. 100 W
D. 500 W

Answer

To find the wattage of the lamp emitting \(10^{20}\) photons of wavelength 660 nm per second, we can follow these steps:

### Step 1: Understand the relationship between power, energy, and time
Power (P) is defined as the energy (E) emitted per unit time (t):
\[ P = \frac{E}{t} \]
Since we are considering the power emitted per second, we can express this as:
\[ P = E \quad \text{(for } t = 1 \text{ second)} \]

### Step 2: Calculate the energy of one photon
The energy of a single photon can be calculated using the formula:
\[ E = h \nu \]
where \(h\) is Planck's constant and \(\nu\) is the frequency of the photon. The frequency can be related to the wavelength (\(\lambda\)) using the equation:
\[ \nu = \frac{c}{\lambda} \]
where \(c\) is the speed of light.

### Step 3: Substitute the frequency into the energy equation
Substituting the expression for frequency into the energy equation gives:
\[ E = h \frac{c}{\lambda} \]

### Step 4: Plug in the values
- Planck's constant \(h = 6.626 \times 10^{-34} \, \text{J s}\)
- Speed of light \(c = 3 \times 10^8 \, \text{m/s}\)
- Wavelength \(\lambda = 660 \, \text{nm} = 660 \times 10^{-9} \, \text{m}\)

Now, substituting these values into the energy equation:
\[ E = \left(6.626 \times 10^{-34}\right) \left(\frac{3 \times 10^8}{660 \times 10^{-9}}\right) \]

### Step 5: Calculate the energy of one photon
Calculating the frequency:
\[ \nu = \frac{3 \times 10^8}{660 \times 10^{-9}} = \frac{3 \times 10^8}{6.6 \times 10^{-7}} \approx 4.545 \times 10^{14} \, \text{Hz} \]
Now, substituting this back to find the energy:
\[ E = 6.626 \times 10^{-34} \times 4.545 \times 10^{14} \approx 3.01 \times 10^{-19} \, \text{J} \]

### Step 6: Calculate the total energy emitted per second
Since \(10^{20}\) photons are emitted per second, the total energy emitted per second (which is the power) is:
\[ P = n \cdot E = 10^{20} \cdot 3.01 \times 10^{-19} \approx 30.1 \, \text{W} \]

### Final Answer
The wattage of the lamp is approximately:
\[ \boxed{30.1 \, \text{W}} \]