To solve the problem, we need to analyze the situation involving the 20W bulb, the wavelength of light, the work function of the metal, and the dimensions of the atoms on the metal surface. Here's a step-by-step solution:
### Step 1: Calculate the total energy emitted by the bulb per second
The power of the bulb is given as \(20W\). This means it emits \(20J\) of energy every second.
### Step 2: Calculate the number of photons emitted per second
The energy of a single photon can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
where:
- \(h\) (Planck's constant) = \(6.626 \times 10^{-34} \, J \cdot s\)
- \(c\) (speed of light) = \(3 \times 10^8 \, m/s\)
- \(\lambda\) (wavelength) = \(5000 \, Å = 5000 \times 10^{-10} \, m\)
Now, substituting the values:
\[
E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{5000 \times 10^{-10}}
\]
Calculating this gives:
\[
E = \frac{1.9878 \times 10^{-25}}{5 \times 10^{-7}} = 3.9756 \times 10^{-19} \, J
\]
### Step 3: Calculate the number of photons emitted per second
Now, we can find the number of photons emitted per second by dividing the total energy emitted by the energy of a single photon:
\[
\text{Number of photons} = \frac{20 \, J}{3.9756 \times 10^{-19} \, J} \approx 5.03 \times 10^{19} \, \text{photons/second}
\]
### Step 4: Calculate the area of the circular disk representing an atom
The area \(A\) of a circular disk can be calculated using the formula:
\[
A = \pi r^2
\]
where \(r = 1.5 \, Å = 1.5 \times 10^{-10} \, m\).
Calculating the area:
\[
A = \pi (1.5 \times 10^{-10})^2 \approx 7.0686 \times 10^{-20} \, m^2
\]
### Step 5: Calculate the number of atoms illuminated by the bulb
To find the number of atoms that can be illuminated by the light, we need to consider the intensity of light at the surface. The intensity \(I\) can be calculated as:
\[
I = \frac{P}{A}
\]
where \(P\) is the power of the bulb and \(A\) is the area over which the light is spread. The area at a distance \(d = 2m\) from the bulb is:
\[
A = 4\pi d^2 = 4\pi (2)^2 = 16\pi \approx 50.27 \, m^2
\]
Now, substituting the values:
\[
I = \frac{20 \, W}{50.27 \, m^2} \approx 0.397 \, W/m^2
\]
### Step 6: Calculate the number of atoms that can absorb the photons
The number of atoms that can be illuminated can be calculated by dividing the intensity by the energy required to free an electron (work function):
\[
\text{Number of atoms} = \frac{I}{\text{Work Function}} = \frac{0.397 \, W/m^2}{2 \times 1.6 \times 10^{-19} \, J} \approx 1.24 \times 10^{18} \, \text{atoms/m}^2
\]
### Step 7: Calculate the total number of atoms illuminated
Finally, we multiply the number of atoms per square meter by the area of the surface:
\[
\text{Total number of atoms} = \text{Number of atoms per m}^2 \times \text{Area}
\]
Assuming the area of the metal surface is \(1 m^2\):
\[
\text{Total number of atoms} \approx 1.24 \times 10^{18} \, \text{atoms}
\]
### Summary
The total number of atoms illuminated by the light from the bulb is approximately \(1.24 \times 10^{18}\).
