Light of wavelength \(300 \times 10^{-9} \mathrm{~m}\) strikes a metal surface with photoelectric work function of \(2.13 \mathrm{eV}\). Find out the kinetic energy of the most energetic photoelectron.

Imagine there's an energetic threshold that electrons need to overcome in order to break free from a metal's surface. This threshold is called the work function, and it's unique to each material. While the work function is often given in electron volts (eV), we need to convert it to joules (J) for consistency with the International System of Units (SI).

To convert eVs to joules, we use the simple conversion factor: 1 eV = 1.602 x 10^-19 J. Thus for our problem, the work function in joules is calculated as: \(2.13 \text{{ eV }} \times 1.602 \times 10^{-19} \text{{ J/eV}}\). Understanding the work function in joules is crucial for solving many photoelectric effect problems.

Every photon, or particle of light, carries a specific amount of energy which depends on its frequency. However, dealing with frequency isn't always practical, so we often use wavelength as it's what we typically know. To find the energy of a photon using its wavelength, we apply a fundamental formula: \(E = \frac{{hc}}{{\lambda}}\), where E is the photon's energy, h is Planck's constant, c is the speed of light, and \lambda is the wavelength of the light.

For our exercise, the photon energy is calculated using the given wavelength of the light and remembering the value of Planck's constant and the speed of light in a vacuum. This equation bridges the concepts of light as both a wave and particle, integral to understanding the photoelectric effect.

A tiny numerical value bridges the macro world we see and the quantum world we don't: Planck's constant (\(h\)), fundamental to quantum physics. It is approximately 6.626 x 10^-34 J*s and is a proportionality factor in the equation that relates photon energy to its frequency. Its relevance extends beyond calculations, anchoring the quantum nature of energy exchange.

In the context of the photoelectric effect, Planck's constant is used along with the speed of light to calculate the photon energy. The incredibly small size of Planck's constant reflects the minute scale of action in the quantum realm and reveals the discrete nature of energy transfer in the form of photons.

When a photon with enough energy strikes a metal surface, it can eject electrons, giving them kinetic energy. In the photoelectric effect, the maximum kinetic energy (\(KE_{max}\)) of an ejected photoelectron is the difference between the energy of the incoming photon and the work function of the metal: \(KE_{max} = E_{photon} - W\), where \(E_{photon}\) is the photon's energy and \(W\) is the work function.

In our problem, after calculating the photon's energy and converting the work function to joules, we deduct the latter from the former to find the photoelectron's kinetic energy. This value tells us how much energy the photoelectron will have after escaping the surface, crucial for understanding the dynamics of the photoelectric effect.