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The concept of the **work function** is integral to understanding the photoelectric effect. The work function refers to the minimum energy required to eject an electron from the surface of a material. This energy depends on the type of material and is often expressed in energy per mole of the substance. For instance, in the context of metals like cesium, the work function determines how much energy a photon must have to knock out an electron. In our example, the work function for cesium is given as \( 188,000 \text{ J/mol} \). To find the energy per photon, we would convert this using Avogadro's number, \( 6.022 \times 10^{23} \text{ mol}^{-1} \), resulting in approximately \( 3.12 \times 10^{-19} \text{ J/photon} \). If photons striking the material have energy below this threshold, no electrons will be ejected regardless of the number of photons present.

Calculating the energy of a photon is a fundamental step in evaluating the photoelectric effect. This is done using the formula: \[ E = \frac{hc}{\lambda} \]where \( E \) is energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), \( c \) is the speed of light \( (3.00 \times 10^{8} \text{ m/s}) \), and \( \lambda \) is the wavelength of the light. When calculating the photon energy for different wavelengths: - For a wavelength of \( 500 \text{ nm} \), the energy comes out to \( 3.975 \times 10^{-19} \text{ J/photon} \). - For \( 650 \text{ nm} \), it's \( 3.06 \times 10^{-19} \text{ J/photon} \). Comparing these energies with the work function of a material helps determine if an electron will be ejected. If the energy per photon is less than the work function, the photon will not have enough power to free an electron.

Wavelength and frequency are closely linked characteristics of light that influence photon energy. The wavelength of light, denoted as \( \lambda \), is the distance between consecutive peaks of a wave and is usually measured in nanometers for visible light. The **frequency** of light, represented by \( u \), is how often these peaks pass a point per second. Wavelength and frequency are inversely related through the formula: \[ c = \lambda u \]where \( c \) is the speed of light. Thus, a shorter wavelength implies a higher frequency and, consequently, higher energy photons when compared to longer wavelengths. In practice, understanding this relationship aids in predicting the behaviour of light when it interacts with materials. For instance, in the exercise, a light of 650 nm results in lower energy compared to a 500 nm light.

**Electron ejection** is the process of dislodging electrons from a material's surface when it is exposed to light. This phenomenon is central to the photoelectric effect. For electron ejection to occur, the energy of incoming photons must exceed the work function of the material. Once the photons successfully transfer enough energy, the electrons absorb it and escape from the surface. There are a few factors that influence this effect:

• Photon Energy: Only photons with energy greater than the work function will cause ejection.
• Wavelength: Shorter wavelengths, corresponding to higher energies, are more effective in causing ejection.
• Intensity: Amplitude affects the number of photons but not the energy per photon.

In this exercise, options with a wavelength of 500 nm effectively eject electrons, as their photon energy surpasses the work function of cesium. However, at 650 nm, too little energy per photon means no electrons are ejected, regardless of the wave amplitude.