If \(\lambda_{0}\) and \(\lambda\) be the threshold wavelength and the wavelength of incident light, the velocity of photo-electrons ejected from the metal surface is : (a) \(\sqrt{\frac{2 h}{m}\left(\lambda_{0}-\lambda\right)}\) (b) \(\sqrt{\frac{2 h c}{m}\left(\lambda_{0}-\lambda\right)}\) (c) \(\sqrt{\frac{2 h c}{m}\left(\frac{\lambda_{0}-\lambda}{\lambda \lambda_{0}}\right)}\) (d) \(\sqrt{\frac{2 h}{m}\left(\frac{1}{\lambda_{0}}-\frac{1}{\lambda}\right)}\)

Understanding the concept of threshold wavelength is crucial for grasping the fundamentals of the photoelectric effect. It represents the specific wavelength of light that is just sufficient to eject electrons from a material surface. In other words, if the light has a wavelength longer than the threshold wavelength, no electrons will be ejected, regardless of the light's intensity.

This critical value, denoted as \( \lambda_0 \), is intimately related to the material's work function - the minimum energy needed to remove an electron from the material. Different materials have different threshold wavelengths because their atomic structures require varying amounts of energy to release an electron.

Significance in Photoelectric Effect

The threshold wavelength marks the boundary between light that can cause the photoelectric effect and light that cannot. As it is tied to the atomic properties of a metal's surface, understanding and determining the threshold wavelength helps in the study of material properties and the design of photoelectric devices.

When it comes to the photoelectric effect, the kinetic energy of photo-electrons is a measure of the energy possessed by the ejected electrons as they leave the metal surface. This kinetic energy is the difference between the energy of the incoming photons and the work function of the material. Precisely, it is given by the formula \( E_{kinetic} = \frac{hc}{\lambda} - \frac{hc}{\lambda_0} \), which represents the energy conversion from photons to ejected electrons.

The Kinetic energy of the electrons will be zero if the photon energy is equal to the work function (when \( \lambda = \lambda_0 \) ). If the photon energy is higher, the excess energy is transferred to the electrons as kinetic energy, causing them to be released at higher velocities.

Quantitative Insights

Quantitatively, the faster the ejected photo-electrons, the more kinetic energy they have. This is significant in various applications, including solar panels and photomultipliers, where the efficiency of the photoelectric process is pivotal. Calculating this energy can help predict the electrons' behavior after they are liberated from the metal surface.

Planck's constant, symbolized by \( h \), is a fundamental constant in physics, particularly quantum mechanics. It signifies the proportionality factor between the frequency of a photon and its energy. The constant has a value of approximately \( 6.62607015 \times 10^{-34} \mathrm{J \cdot s} \).

Named after physicist Max Planck, who founded quantum theory, this constant is instrumental in quantifying the discreetness of energy levels in quantum systems. Planck's constant bridges the classical and quantum worlds, leading to its vital role in formulas related to the photoelectric effect, atomic structure, and the energy levels of atoms.

Role in Photoelectric Effect

In the context of the photoelectric effect, Planck's constant is used to calculate the energy of photons involved in ejecting electrons from a metal surface. It elucidates the particle-like properties of light, which a century ago was viewed purely as a wave phenomenon. The precise knowledge of \( h \), therefore, is pivotal for making accurate predictions in light-matter interactions, like those observed in the photoelectric effect.