Let's solve the given problem step-by-step.
### Problem:
We are given:
- The wavelength of the incident light: [tex]\( \lambda = 400 \)[/tex] nm
- The work function of the metal: [tex]\( \phi = 2.3 \)[/tex] eV
We need to find:
1. The wavelength in meters.
2. The energy of a photon with this wavelength.
3. The maximum kinetic energy of the photoelectrons ejected.
### Step 1: Convert the Wavelength to Meters
The wavelength is given in nanometers. We need to convert this to meters.
[tex]\[
\lambda = 400 \text{ nm} = 400 \times 10^{-9} \text{ m} = 4.0 \times 10^{-7} \text{ m}
\][/tex]
### Step 2: Calculate the Energy of a Photon
The energy of a photon can be calculated using the formula:
[tex]\[
E = \frac{h \cdot c}{\lambda}
\][/tex]
Where:
- [tex]\( h \)[/tex] is Planck's constant, approximately [tex]\( 4.135667696 \times 10^{-15} \)[/tex] eV·s
- [tex]\( c \)[/tex] is the speed of light, approximately [tex]\( 3.0 \times 10^8 \)[/tex] m/s
Substituting the values, we get:
[tex]\[
E = \frac{4.135667696 \times 10^{-15} \text{ eV·s} \times 3.0 \times 10^8 \text{ m/s}}{4.0 \times 10^{-7} \text{ m}}
\][/tex]
Calculating this gives us the photon energy:
[tex]\[
E \approx 3.101750772 \text{ eV}
\][/tex]
### Step 3: Calculate the Maximum Kinetic Energy of the Ejected Photoelectron
The maximum kinetic energy ([tex]\( K_{\text{max}} \)[/tex]) of the photoelectrons can be found using the photoelectric equation:
[tex]\[
K_{\text{max}} = E - \phi
\][/tex]
Where:
- [tex]\( \phi \)[/tex] is the work function (2.3 eV)
Substituting the values:
[tex]\[
K_{\text{max}} = 3.101750772 \text{ eV} - 2.3 \text{ eV}
\][/tex]
Calculating this gives us:
[tex]\[
K_{\text{max}} \approx 0.801750772 \text{ eV}
\][/tex]
### Summary
In summary, given a wavelength of 400 nm, we have:
1. Converted the wavelength to meters: [tex]\( 4.0 \times 10^{-7} \)[/tex] meters.
2. Calculated the energy of a photon with this wavelength: approximately [tex]\( 3.101750772 \)[/tex] eV.
3. Determined the maximum kinetic energy of the photoelectrons considering the work function of the metal: approximately [tex]\( 0.801750772 \)[/tex] eV.
These calculations are crucial in understanding the photoelectric effect and the behavior of light interacting with materials.
