1. Photoelectric effect

• The minimum frequency of light that can cause the photoelectric effect in a metal with a work function of 2.0 eV can be obtained from Einstein’s photoelectric equation:

$$hf_0 = \phi$$

where $$hf_0$$ is the energy of the photon, $$\phi$$ is the work function, and $$h$$ is Planck’s constant (6.626 × 10^-34 Js). Rearranging the equation and substituting values:

$$f_0 = \frac{\phi}{h} = \frac{2.0 \text{ eV}}{(6.626 \times 10^{-34} \text{ Js})}$$

$$f_0 = 4.81 \times 10^{14} \text{ Hz}$$

To calculate the corresponding wavelength, we can use the speed of light (c = 3.0 × 10^8 m/s) and the equation:

$$\lambda_0=\frac{c} { f_0}$$

$$\lambda = 4.0 \times 10^{-7}\ m=624 \text{ nm}$$

• If the same light incident on a metal with a work function of 3.0 eV, the maximum kinetic energy (Kmax) of the emitted electrons can be determined using Einstein’s photoelectric equation:

$$K_{max} = hf - \phi$$

Substituting values:

$$K_{max} = (6.626 \times 10^{-34} \text{ Js})(4.81 \times 10^{14} \text{ Hz}) - 3.0 \text{ eV}$$

$$K_{max} = 1.83 \times 10^{-19} \text{ J}$$

• If the wavelength of the incident light is 400 nm, the photon energy can be calculated:

$$E = h f = \frac{hc}{\lambda}$$

where $$\lambda$$ is the wavelength of light, $$h$$ is Planck’s constant (6.626 × 10^-34 Js), and $$c$$ is the speed of light (3.0 × 10^8 m/s):

$$E = \frac{(6.626 \times 10^{-34} \text{ Js})(3.0 \times 10^8 \text{ m/s})}{400 \times 10^{-9} \text{ m}}$$

$$E = 4.97 \times 10^{-19} \text{ J}$$

Then, using Einstein’s photoelectric equation:

$$K_{max} = E - \phi = 4.97 \times 10^{-19} \text{ J} - 2.0 \text{ eV}$$

$$K_{max} = 2.97 \times 10^{-19} \text{ J}$$

The stopping potential (V) for the photoelectrons can be calculated using the formula:

$$eV = K_{max}$$

where $$e$$ is the charge of an electron (1.602 × 10^-19 C):

$$V = \frac{K_{max}}{e} = \frac{2.97 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ C}}$$

$$V = 1.85 \text{ V}$$

2. Photoelectric effect and the particle nature of light

• Einstein’s photoelectric equation: $$hf= \phi +K_{max}$$

If light were a continuous wave, then increasing the intensity (or amplitude) of the light would increase the wave energy. According to the classical wave theory, this should result in an increase in the kinetic energy of the emitted electrons.

However, experiments showed that the kinetic energy of the emitted electrons depends only on the frequency (or energy) of the incident light, and not on its intensity.

This observation is consistent with the particle nature of light, as proposed by Einstein in the photoelectric effect theory.

Photons, or quanta of light, carry a discrete amount of energy (hf) proportional to their frequency. When a photon strikes an electron in the metal, it can transfer its energy to the electron, enabling it to escape from the metal’s surface. This process is independent of the light’s intensity because what matters is the energy carried by each individual photon, not the overall intensity of the light beam.

• Maximum kinetic energy of the emitted electron:

$$K_{max} = hf - \phi$$

Substituting values:

$$K_{max} = (6.626 \times 10^{-34} \text{ Js})(6.626 \times 10^{14} \text{ Hz}) - 2.0 \text{ eV}$$

$$K_{max} = 0.663 \times 10^{-19} \text{ J}$$

• Stopping potential:

$$eV = K_{max}$$

Substituting values:

$$V = \frac{K_{max}}{e} = \frac{0.663 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ C}}$$

$$V = 0.414 \text{ V}$$

3. Applications of the photoelectric effect

• Working principle of a solar cell:

A solar cell is a device that converts light energy into electrical energy. It is based on the photoelectric effect. When sunlight strikes the semiconductor material in a solar cell, it generates electron-hole pairs. The electrons are then separated from the holes and directed towards the positive terminal of the cell, while the holes move towards the negative terminal. This creates a flow of electric current.

• Measurement of work function:

The photoelectric effect can be used to measure the work function of a metal. The experimental setup involves illuminating the metal surface with light of varying wavelengths and measuring the corresponding stopping potential. By plotting the stopping potential versus the frequency of the incident light, a straight line is obtained. The work function can be determined from the slope of this line.

4. Advanced concepts in the photoelectric effect

• Quantum efficiency:

Quantum efficiency (QE) is a measure of the efficiency of a photoelectric material in converting incident photons into electric charge carriers (usually electrons). It is defined as the ratio of the number of charge carriers generated to the number of incident photons. The QE is affected by several factors, such as the band gap energy, absorption coefficient, and surface recombination of the material. It can be improved by optimizing these factors, such as using materials with a suitable band gap and employing anti-reflection coatings to reduce surface recombination.

• Einstein-de Haas relation:

The Einstein-de Haas relation expresses the relationship between the magnetic moment induced in a material by the photoelectric effect and the charge and mass of the emitted electrons.

$$M=\frac{Neh}{2m}$$

Here M represents the magnetic moment caused by the photoelectric effect, N is the total number of emitted electrons and e is the charge of each electron. The constant h represents the Planck’s constant, while m denotes the mass of each emitted electron.

• Surface plasmons:

Surface plasmons are collective oscillations of electrons at the interface between a metal and a dielectric material. They can enhance the photoelectric effect by increasing the absorption of light and confining it to a smaller area. This can improve the efficiency of photoelectric devices.

• Hot-carrier photovoltaics:

Hot-carrier photovoltaics is a concept that utilizes the high-energy electrons generated by the photoelectric effect for efficient solar energy conversion. By extracting and utilizing the energy of these ‘hot’ electrons, beyond the traditional bandgap energy of the semiconductor, it is possible to enhance the overall efficiency of solar cells. This approach involves designing materials and device structures that enable efficient collection and utilization of hot carriers.