Sure, let's go through the problem step-by-step to find the wavelength of the electromagnetic radiation that struck the surface and caused the ejection of photoelectrons.
### Given Data:
1. Work Function, [tex]\( \phi \)[/tex]:
[tex]\[
\phi = 4.60 \times 10^{-19} \, \text{J}
\][/tex]
2. Maximum Speed of Emitted Photoelectrons, [tex]\( v_{\text{max}} \)[/tex]:
[tex]\[
v_{\text{max}} = 2.33 \times 10^6 \, \text{m/s}
\][/tex]
3. Planck's Constant, [tex]\( h \)[/tex]:
[tex]\[
h = 6.62607015 \times 10^{-34} \, \text{J} \cdot \text{s}
\][/tex]
4. Speed of Light, [tex]\( c \)[/tex]:
[tex]\[
c = 3.0 \times 10^8 \, \text{m/s}
\][/tex]
5. Mass of an Electron, [tex]\( m_e \)[/tex]:
[tex]\[
m_e = 9.10938356 \times 10^{-31} \, \text{kg}
\][/tex]
### Step-by-Step Solution:
Step 1: Calculate Kinetic Energy of Photoelectrons
The kinetic energy [tex]\( KE \)[/tex] of the photoelectrons can be calculated using their maximum speed. The formula for kinetic energy is:
[tex]\[
KE = \frac{1}{2} m_e v_{\text{max}}^2
\][/tex]
Substituting the given values:
[tex]\[
KE = \frac{1}{2} \times 9.10938356 \times 10^{-31} \, \text{kg} \times (2.33 \times 10^6 \, \text{m/s})^2
\][/tex]
[tex]\[
KE = 2.4726966204442 \times 10^{-18} \, \text{J}
\][/tex]
Step 2: Calculate Total Energy of Incident Photons
The total energy [tex]\( E \)[/tex] of the incident photons is the sum of the work function [tex]\( \phi \)[/tex] and the kinetic energy [tex]\( KE \)[/tex]:
[tex]\[
E = \phi + KE
\][/tex]
Substituting the given and calculated values:
[tex]\[
E = 4.60 \times 10^{-19} \, \text{J} + 2.4726966204442 \times 10^{-18} \, \text{J}
\][/tex]
[tex]\[
E = 2.9326966204441997 \times 10^{-18} \, \text{J}
\][/tex]
Step 3: Calculate Wavelength of the Incident Radiation
The wavelength [tex]\( \lambda \)[/tex] of the incident radiation can be calculated using the energy of the photons. The relationship between the energy [tex]\( E \)[/tex] of a photon and its wavelength [tex]\( \lambda \)[/tex] is given by the equation:
[tex]\[
\lambda = \frac{h \cdot c}{E}
\][/tex]
Substituting the given and calculated values:
[tex]\[
\lambda = \frac{6.62607015 \times 10^{-34} \, \text{J} \cdot \text{s} \times 3.0 \times 10^8 \, \text{m/s}}{2.9326966204441997 \times 10^{-18} \, \text{J}}
\][/tex]
[tex]\[
\lambda = 6.778133923374983 \times 10^{-8} \, \text{m}
\][/tex]
### Conclusion
The wavelength of the electromagnetic radiation that struck the surface and caused the ejection of the photoelectrons is:
[tex]\[
\lambda = 6.778133923374983 \times 10^{-8} \, \text{m}
\][/tex]
