The minimum energy required for the emission of photoelectron from the surface of a metal is \(4.95 \times 10^{-19} \mathrm{~J}\). Calculate the critical frequency and the corresponding wavelength of the photon required to eject the electron. \(h=6.6 \times 10^{-34} \mathrm{~J} \mathrm{sec}\).

The work function of a material is the minimum energy required to remove an electron from the surface of a metal. It's a crucial concept in understanding the photoelectric effect, where light is used to eject electrons from a material. This energy threshold is represented by the equation \( E = h u \), where \( E \) is the work function (energy), \( h \) is Planck’s constant, and \( u \) is the critical frequency—the minimum frequency of light required to eject an electron. In simple terms, if the photon's energy is less than the work function, no electrons will be emitted regardless of the intensity of the light. On the other hand, if it meets or exceeds this energy, electrons will be released, with the excess energy appearing as the kinetic energy of the electrons. Understanding this equation allows students to calculate the threshold at which photoelectrons begin to be emitted, providing a foundational understanding of the interactive nature between light and matter.

The critical frequency is directly related to the work function equation, as it is the frequency at which the energy of an incident photon equals the work function. To find this, you rearrange the work function equation to solve for frequency: \( u = \frac{E}{h} \), where \( u \) is the frequency, \( E \) is the work function or the minimum energy, and \( h \) is Planck's constant. This formula is essential in determining whether a photon has enough energy to eject an electron from a metal surface. Knowing how to perform this calculation helps students predict whether a photoelectric effect will be observed for a given frequency of light and understand the quantized nature of photon interactions.

The wavelength of a photon is tied to its frequency through the universal speed of light equation: \( c = \lambda u \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency. Once the critical frequency is known, determining the wavelength is straightforward: \( \lambda = \frac{c}{u} \). This relationship shows that as frequency increases, wavelength decreases, and vice versa. An understanding of how to determine photon wavelength allows you to further explore the characteristics of light and the electromagnetic spectrum. For instance, considering the wavelength helps in explaining the color of light since different wavelengths correspond to different colors visible to the human eye.

Planck's constant \( h \) is a fundamental quantity in quantum mechanics. It represents the proportionality constant between the energy \( E \) of a photon and the frequency \( u \) of its associated electromagnetic wave, as expressed in the equation \( E = hu \). Planck's constant enables us to quantify the discrete packets of energy carried by photons and is pivotal in calculations related to the photoelectric effect. Its application in problems provides students with practical insights into the quantum nature of light and how energy quantization influences physical phenomena on the atomic and sub-atomic scales. By integrating Planck's constant into problems like the one above, students can bridge the gap between theoretical concepts of quantum physics and their tangible outcomes in experimental observations.