If 5% of the energy supplied to a bulb is irradiated as visible light, how many quanta are emitted per second by a 100 W lamp? Assume wavelength of visible light as `5.6xx10^(-5)` cm.

Options

A. `1.4xx10^(19)`
B. `3xx10^(3)`
C. `1.4xx10^(-19)`
D. `3xx10^(4)`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the energy radiated as visible light
Given that 5% of the energy supplied to a 100 W bulb is radiated as visible light, we can calculate the energy radiated as follows:

\[ \text{Energy radiated (E)} = 5\% \times 100 \text{ W} = 0.05 \times 100 = 5 \text{ J/s} \]

### Step 2: Convert the wavelength from cm to meters
The wavelength of visible light is given as \(5.6 \times 10^{-5}\) cm. We need to convert this to meters:

\[ \text{Wavelength} (\lambda) = 5.6 \times 10^{-5} \text{ cm} = 5.6 \times 10^{-5} \times 10^{-2} \text{ m} = 5.6 \times 10^{-7} \text{ m} \]

### Step 3: Use the formula to find the number of quanta (photons) emitted per second
The energy of light can be expressed in terms of the number of photons (n) using the formula:

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

Where:
- \(E\) is the energy radiated (5 J/s)
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \text{ J s}\))
- \(c\) is the speed of light (\(3 \times 10^{8} \text{ m/s}\))
- \(\lambda\) is the wavelength of light (\(5.6 \times 10^{-7} \text{ m}\))

Rearranging the formula to solve for \(n\):

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

### Step 4: Substitute the values into the equation
Now we can substitute the values we have into the equation:

\[ n = \frac{5 \text{ J/s} \cdot 5.6 \times 10^{-7} \text{ m}}{(6.63 \times 10^{-34} \text{ J s}) \cdot (3 \times 10^{8} \text{ m/s})} \]

### Step 5: Calculate the value of n
Calculating the denominator:

\[ hc = (6.63 \times 10^{-34}) \cdot (3 \times 10^{8}) = 1.989 \times 10^{-25} \text{ J m} \]

Now substituting back into the equation for \(n\):

\[ n = \frac{5 \cdot 5.6 \times 10^{-7}}{1.989 \times 10^{-25}} \]

Calculating the numerator:

\[ 5 \cdot 5.6 \times 10^{-7} = 2.8 \times 10^{-6} \]

Now calculating \(n\):

\[ n = \frac{2.8 \times 10^{-6}}{1.989 \times 10^{-25}} \approx 1.41 \times 10^{19} \text{ photons/s} \]

### Final Answer
Thus, the number of quanta (photons) emitted per second by the 100 W lamp is approximately:

\[ n \approx 1.4 \times 10^{19} \text{ photons/s} \]