To solve the problem, we will follow these steps:
### Step 1: Calculate the Power of the Incident Light
The power \( P \) of the incident light can be calculated using the formula:
\[
P = I \times A
\]
where \( I \) is the intensity of the light and \( A \) is the area of the photocell.
Given:
- Intensity \( I = 10^{-5} \, \text{W/m}^2 \)
- Area \( A = 2 \, \text{cm}^2 = 2 \times 10^{-4} \, \text{m}^2 \)
Calculating the power:
\[
P = 10^{-5} \, \text{W/m}^2 \times 2 \times 10^{-4} \, \text{m}^2 = 2 \times 10^{-9} \, \text{W}
\]
### Step 2: Calculate the Number of Atoms in the Top 5 Layers
To find the number of atoms \( N' \) in the top 5 layers of sodium, we use the formula:
\[
N' = n \times \frac{A}{A_{\text{atom}}}
\]
where \( n \) is the number of layers, \( A \) is the area, and \( A_{\text{atom}} \) is the effective area of a sodium atom.
Given:
- Number of layers \( n = 5 \)
- Effective atomic area of sodium \( A_{\text{atom}} = 10^{-20} \, \text{m}^2 \)
Calculating the number of atoms:
\[
N' = 5 \times \frac{2 \times 10^{-4} \, \text{m}^2}{10^{-20} \, \text{m}^2} = 5 \times 2 \times 10^{16} = 10^{17}
\]
### Step 3: Calculate the Energy Absorbed per Second per Electron
The energy absorbed per second per electron \( E \) is given by:
\[
E = \frac{P}{N'}
\]
Substituting the values:
\[
E = \frac{2 \times 10^{-9} \, \text{W}}{10^{17}} = 2 \times 10^{-26} \, \text{J/s}
\]
### Step 4: Convert Work Function to Joules
The work function \( W \) is given in electron volts (eV). We need to convert it to joules:
\[
W = 2 \, \text{eV} = 2 \times 1.6 \times 10^{-19} \, \text{J} = 3.2 \times 10^{-19} \, \text{J}
\]
### Step 5: Calculate the Time Required for Photoelectric Emission
The time \( T \) required for photoelectric emission can be calculated using:
\[
T = \frac{E}{W}
\]
Substituting the values:
\[
T = \frac{2 \times 10^{-26} \, \text{J/s}}{3.2 \times 10^{-19} \, \text{J}} = 6.25 \times 10^{-8} \, \text{s}
\]
### Step 6: Convert Time to Years
To convert seconds to years, we use the conversion factor \( 1 \, \text{year} = 3.154 \times 10^7 \, \text{s} \):
\[
T_{\text{years}} = \frac{6.25 \times 10^{-8} \, \text{s}}{3.154 \times 10^7 \, \text{s/year}} \approx 1.98 \times 10^{-15} \, \text{years}
\]
### Final Answer
The estimated time required for photoelectric emission is approximately \( 1.98 \times 10^{-15} \, \text{years} \).
