To solve the problem, we need to find the rate at which photons strike the surface, denoted as \( K \times 10^{13} \) photons/s. We will follow these steps:
### Step 1: Convert Wavelength to Meters
The wavelength \( \lambda \) is given as 3000 Å (angstroms). We need to convert this to meters:
\[
\lambda = 3000 \, \text{Å} = 3000 \times 10^{-10} \, \text{m} = 3.0 \times 10^{-7} \, \text{m}
\]
**Hint:** Remember that 1 Å = \( 10^{-10} \) m.
### Step 2: Calculate the Energy of One Photon
The energy \( E \) of a single photon can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
Where:
- \( h = 6.626 \times 10^{-34} \, \text{J s} \) (Planck's constant)
- \( c = 3.0 \times 10^{8} \, \text{m/s} \) (speed of light)
Substituting the values:
\[
E = \frac{(6.626 \times 10^{-34}) \times (3.0 \times 10^{8})}{3.0 \times 10^{-7}}
\]
Calculating this gives:
\[
E = 6.626 \times 10^{-19} \, \text{J}
\]
**Hint:** Use the values of \( h \) and \( c \) carefully, and ensure units are consistent.
### Step 3: Calculate the Area in Square Meters
The area \( A \) is given as \( 4 \, \text{cm}^2 \). We need to convert this to square meters:
\[
A = 4 \, \text{cm}^2 = 4 \times 10^{-4} \, \text{m}^2
\]
**Hint:** Remember that \( 1 \, \text{cm}^2 = 10^{-4} \, \text{m}^2 \).
### Step 4: Calculate the Power Incident on the Surface
The intensity \( I \) is given as \( 15 \times 10^{-2} \, \text{W/m}^2 \). The power \( P \) incident on the surface can be calculated using:
\[
P = I \times A
\]
Substituting the values:
\[
P = (15 \times 10^{-2}) \times (4 \times 10^{-4}) = 6.0 \times 10^{-5} \, \text{W}
\]
**Hint:** Power is the product of intensity and area.
### Step 5: Calculate the Number of Photons Striking the Surface per Second
The number of photons \( n \) striking the surface per second can be calculated using:
\[
n = \frac{P}{E}
\]
Substituting the values:
\[
n = \frac{6.0 \times 10^{-5}}{6.626 \times 10^{-19}} \approx 9.05 \times 10^{13} \, \text{photons/s}
\]
**Hint:** Make sure to divide the power by the energy of one photon to find the number of photons.
### Step 6: Determine the Value of \( K \)
From the calculation, we have:
\[
n \approx 9.05 \times 10^{13} \, \text{photons/s} = K \times 10^{13}
\]
Thus, \( K \) is:
\[
K \approx 9.05
\]
**Final Answer:** The value of \( K \) is approximately \( 9.05 \).
