To solve the problem, we will use the photoelectric effect principles.

Understanding the Photoelectric Effect
In the photoelectric effect, when light falls on a metal surface, it can cause the emission of electrons. The energy of the incoming photons is compared to the work function of the metal to determine if electrons are emitted and their kinetic energy.

Given Data
- **Wavelength (λ)**: 400 nm (which is \(400 \times 10^{-9}\) m)
- **Work function (Φ)**: 2 eV

Calculating Energy of the Photon
The energy of a photon (E) can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
Where:
- \(h\) = Planck's constant \( = 6.626 \times 10^{-34} \text{ J·s}\)
- \(c\) = Speed of light \( = 3 \times 10^{8} \text{ m/s}\)
Converting the energy to electron volts (1 eV = \(1.6 \times 10^{-19}\) J):
1. Calculate \(E\):
\[
E = \frac{(6.626 \times 10^{-34})(3 \times 10^{8})}{400 \times 10^{-9}} \approx 4.97 \times 10^{-19} \text{ J}
\]
Converting to eV:
\[
E \approx \frac{4.97 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 3.11 \text{ eV}
\]

Finding the Maximum Kinetic Energy (KE_max)
The maximum kinetic energy of the emitted electrons is given by:
\[
KE_{max} = E - \Phi
\]
Where \(E\) is the energy of the photon and \(Φ\) is the work function.
2. Calculate \(KE_{max}\):
\[
KE_{max} = 3.11 \text{ eV} - 2 \text{ eV} = 1.11 \text{ eV}
\]

Determining the Threshold Wavelength (λ₀)
The threshold wavelength (λ₀) can be found using:
\[
\lambda_0 = \frac{hc}{\Phi}
\]
3. Calculate \(λ_0\):
\[
\lambda_0 = \frac{(6.626 \times 10^{-34})(3 \times 10^{8})}{2 \times 1.6 \times 10^{-19}} \approx 6.20 \times 10^{-7} \text{ m} \approx 620 \text{ nm}
\]

Summary
- **KE_max**: 1.11 eV
- **Threshold Wavelength (λ₀)**: 620 nm
This analysis demonstrates how light interacts with materials at a quantum level, providing insights into energy transfer and electron dynamics.