To solve the problem step by step, we will follow these calculations:
### Step 1: Calculate the energy of a single photon
The energy of a photon can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
Where:
- \( h \) is Planck's constant multiplied by the speed of light, given as \( hc = 1240 \, \text{eV-nm} \)
- \( \lambda \) is the wavelength of the light, given as \( 400 \, \text{nm} \)
Substituting the values:
\[
E = \frac{1240 \, \text{eV-nm}}{400 \, \text{nm}} = 3.1 \, \text{eV}
\]
### Step 2: Convert the power of the light beam to energy per second
The power of the light beam is given as \( 1.55 \, \text{mW} \). To convert this to watts:
\[
1.55 \, \text{mW} = 1.55 \times 10^{-3} \, \text{W}
\]
### Step 3: Calculate the energy of one photon in joules
To convert the energy of one photon from eV to joules, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{C} \):
\[
E = 3.1 \, \text{eV} = 3.1 \times 1.6 \times 10^{-19} \, \text{J} = 4.96 \times 10^{-19} \, \text{J}
\]
### Step 4: Calculate the number of photons incident per second
The number of photons incident per second can be calculated using the formula:
\[
\text{Number of photons} = \frac{\text{Power}}{\text{Energy of one photon}}
\]
Substituting the values:
\[
\text{Number of photons} = \frac{1.55 \times 10^{-3} \, \text{W}}{4.96 \times 10^{-19} \, \text{J}} \approx 3.13 \times 10^{15} \, \text{photons/s}
\]
### Step 5: Calculate the effective number of photons producing photoelectrons
Since only 10% of the incident photons produce photoelectrons:
\[
\text{Effective photons} = 0.10 \times 3.13 \times 10^{15} \approx 3.13 \times 10^{14} \, \text{photons/s}
\]
### Step 6: Calculate the current due to the produced photoelectrons
The current \( I \) can be calculated using the formula:
\[
I = \text{Number of electrons per second} \times \text{Charge of one electron}
\]
Where the charge of one electron is \( e = 1.6 \times 10^{-19} \, \text{C} \):
\[
I = 3.13 \times 10^{14} \, \text{electrons/s} \times 1.6 \times 10^{-19} \, \text{C} \approx 5.008 \times 10^{-5} \, \text{A}
\]
### Step 7: Convert the current to microamperes
To convert from amperes to microamperes:
\[
I \approx 5.008 \times 10^{-5} \, \text{A} = 50.08 \, \mu\text{A}
\]
### Final Answer
Thus, the current due to the photoelectrons is approximately:
\[
\text{Current} \approx 50 \, \mu\text{A}
\]
