A photon of wavelength \(4 \times 10^{-7} \mathrm{~m}\) strikes on metal surface, the work function of the metal being \(2.13 \mathrm{eV}\). Calculate : (i) the energy of the photon (eV), (ii) the kinetic energy of the emission, and (iii) the velocity of the photoelectron \(\left(1 \mathrm{eV}=1.6020 \times 10^{-19} \mathrm{~J}\right)\).

Understanding the concept of photon energy calculation involves a fundamental equation in physics: the relationship between a photon's energy, its wavelength, and Planck's constant. This is expressed by the formula:
\[E = \frac{hc}{\lambda}\]
where \(E\) is the energy of the photon, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J\cdot s}\)), \(c\) is the speed of light (\(3 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the photon's wavelength. After computing the energy in joules, it's often necessary to convert it to electron volts (eV), which is a more convenient unit in atomic-scale interactions. This conversion makes use of the fact that \(1 \mathrm{eV}\) is equivalent to \(1.6020 \times 10^{-19} \mathrm{J}\). By substituting in the given photon wavelength, we can compute the exact energy of the photon striking the metal surface.

In the photoelectric effect, the work function is a critical component to consider. It represents the minimum energy needed to eject an electron from the surface of a metal. This threshold energy varies for different materials and is a fixed property for each.
When a photon with energy greater than the work function hits the metal, it imparts its energy to an electron. If this energy exceeds the work function (\(\phi\)), what is left is converted into the kinetic energy (KE) of the emitted electron. Expressed mathematically, this is:
\[KE = E - \phi\]
where \(E\) is the energy of the incoming photon and \(\phi\) is the metal's work function in electron volts (eV). In the case where the photon's energy is less than the work function, no electrons are emitted as the required threshold isn't met. This concept explains why different metals will emit photoelectrons with varying efficiencies when illuminated with light of the same frequency.

The kinetic energy (KE) of emitted electrons during the photoelectric effect is essentially the energy surplus once the work function threshold is managed. After an electron absorbs a photon's energy and overcomes the binding energy as dictated by the work function, the left-over energy manifests as the electron's kinetic energy.

Using the formula from our previously described work function concept, we deduce that KE is the difference between the energy of the incident photon and the work function. If the photon's energy is precisely equal to the work function, the electron will be emitted with zero kinetic energy—meaning it won't be moving. Any additional energy is seen in the velocity of the ejected electron. This is essential for understanding the dynamics of photoelectrons as they are emitted from various materials under different kinds of light.

Calculating the velocity of photoelectrons requires us to connect the dots between their kinetic energy and mass using classical mechanics. The kinetic energy imparted to photoelectrons can be related to their velocity using the equation:
\[KE = \frac{1}{2}mv^2\]
where \(m\) is the mass of the electron (\(9.109 \times 10^{-31} \mathrm{kg}\)) and \(v\) is its velocity. By rearranging this formula to solve for \(v\), we can derive the electrons' velocity after they're ejected. Firstly, we solve for Kinetic Energy (KE) using the excess photon energy after overcoming the work function. Then, using the known mass of the electron, we can solve for \(v\) by extracting the square root of the rearranged kinetic energy equation. It's important to note that while this approach uses classical physics, once velocities get very high, closer to the speed of light, relativistic effects come into play for which a different calculation method would be needed. For typical photoelectric effect problems in a high school or early college setting, this classical approach is sufficient.