Calculate the number of photons emitted by a `100 W` yellow lamp in `1.0 s`. Take the wavelength of yellow light as `560 nm` and assume `100` percent efficiency.

Options

A. `6.8xx10^(20)`
B. `4xx10^(12)`
C. `4xx10^(20)`
D. `2.8xx10^(20)`

Explanation

To solve the problem of calculating the number of photons emitted by a 100 W yellow lamp in 1.0 s, we will 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). Therefore, we can express energy as:
\[ E = P \times t \]

### Step 2: Calculate the energy emitted in 1 second
Given:
- Power (P) = 100 W
- Time (t) = 1.0 s

Using the formula from Step 1:
\[ E = 100 \, \text{W} \times 1.0 \, \text{s} = 100 \, \text{J} \]

### Step 3: Relate energy to the number of photons
The energy of a single photon can be expressed using the formula:
\[ E = n \cdot h \cdot \nu \]
Where:
- \( n \) = number of photons
- \( h \) = Planck's constant \( (6.626 \times 10^{-34} \, \text{J s}) \)
- \( \nu \) = frequency of the light

### Step 4: Convert wavelength to frequency
The frequency (\( \nu \)) can be calculated using the speed of light (\( c \)) and the wavelength (\( \lambda \)):
\[ \nu = \frac{c}{\lambda} \]
Where:
- \( c \) = speed of light \( (3.0 \times 10^8 \, \text{m/s}) \)
- \( \lambda \) = wavelength of yellow light \( (560 \, \text{nm} = 560 \times 10^{-9} \, \text{m}) \)

Calculating frequency:
\[ \nu = \frac{3.0 \times 10^8 \, \text{m/s}}{560 \times 10^{-9} \, \text{m}} \]
\[ \nu \approx 5.36 \times 10^{14} \, \text{Hz} \]

### Step 5: Substitute into the energy formula
Now, we can express the total energy in terms of the number of photons:
\[ E = n \cdot h \cdot \nu \]
Rearranging gives:
\[ n = \frac{E}{h \cdot \nu} \]

### Step 6: Substitute known values
Substituting the values we have:
- \( E = 100 \, \text{J} \)
- \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- \( \nu \approx 5.36 \times 10^{14} \, \text{Hz} \)

Calculating the number of photons:
\[ n = \frac{100 \, \text{J}}{(6.626 \times 10^{-34} \, \text{J s}) \cdot (5.36 \times 10^{14} \, \text{Hz})} \]

### Step 7: Perform the calculation
Calculating the denominator:
\[ h \cdot \nu \approx 6.626 \times 10^{-34} \times 5.36 \times 10^{14} \approx 3.55 \times 10^{-19} \, \text{J} \]

Now substituting back:
\[ n \approx \frac{100}{3.55 \times 10^{-19}} \approx 2.82 \times 10^{20} \]

### Final Answer
The number of photons emitted by the 100 W yellow lamp in 1.0 s is approximately:
\[ n \approx 2.82 \times 10^{20} \]