Calculate the current produced in a small germanium plate of area \(1 \mathrm{~cm}^{2}\) and of thickness \(03 \mathrm{~mm}\), when a potential difference of 2 volts is apphed across its faces. The intrinsic carrier concentration of geranium is \(2 \times 10^{19} / \mathrm{m}^{3}\) and the mobilities of electron and holes are \(0.36\) and \(0.17 \mathrm{~m}^{2} \mathrm{~V}^{-1} \mathrm{~s}^{-1}\), respectively.

Ohm's Law is fundamental to understanding electrical circuits and conductivity. It states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. The formula can be written as: \( V = IR \). For this exercise, the germanium plate is subjected to a potential difference (voltage) of 2 volts, and we later calculate the resistance. By rearranging Ohm's Law to solve for current, we use the formula: \( I = \frac{V}{R} \). Ohm's Law helps in determining the flow of electric current when the resistance and voltage are known, making it a crucial concept in semiconductor physics.

Electron mobility (\( \mu_e \)) refers to how quickly an electron can move through a semiconductor material when subjected to an electric field. It is measured in \( \text{m}^2/\text{V s} \). In the given problem, the electron mobility for germanium is specified as \( 0.36 \text{m}^2/\text{V s} \). Higher electron mobility means that electrons can move more efficiently through the material, enhancing its conductivity. Electron mobility plays a significant role in determining the overall conductivity of an intrinsic semiconductor, as seen in the earlier conductivity formula: \( \sigma = e (n \mu_e + p \mu_h) \).

Hole mobility ( \( \mu_h \)) represents how quickly holes, which are the absence of electrons acting as positive charge carriers, can move through a semiconductor. Like electron mobility, it is also measured in \( \text{m}^2/\text{V s} \). In this example, the hole mobility for germanium is given as \( 0.17 \text{m}^2/\text{V s} \). While electrons are the negative charge carriers, holes are considered positive charge carriers and contribute to the current flow in the opposite direction of electrons. Hole mobility is essential in understanding the behavior of intrinsic semiconductors, and it is integrated into the expression for conductivity.

Intrinsic carrier concentration (\( n_i \)) refers to the number of charge carriers (free electrons and holes) in a semiconductor material in its pure, undoped state. In germanium, the intrinsic carrier concentration is \( 2 \times 10^{19} \ \text{m}^{-3} \). This value represents an equilibrium number of electrons in the conduction band equal to the number of holes in the valence band. The importance of intrinsic carrier concentration lies in its role in determining the electrical properties of the material. In the given formula for conductivity: \( \sigma = e (n \mu_e + p \mu_h) \), \( n \) and \( p \) are equal to the intrinsic carrier concentration.

Conductivity (\( \sigma \)) measures how well a material can conduct electric current. For intrinsic semiconductors, conductivity is expressed as: \( \sigma = e (n \mu_e + p \mu_h) \). In the exercise, germanium’s conductivity was calculated by substituting the given values of electron charge ( \( e \approx 1.6 \times 10^{-19} \ \text{C} \)), intrinsic carrier concentration (\( n = p = 2 \times 10^{19} \ \text{m}^{-3} \)), electron mobility ( \( \mu_e = 0.36 \ \text{m}^2/\text{V s} \)), and hole mobility ( \( \mu_h = 0.17 \ \text{m}^2/\text{V s} \)). Plugging these into the formula and simplifying, we obtained the conductivity of germanium as \( 1.696 \ \text{S} \ \text{m}^{-1} \). This value helped us further calculate resistance and finally the current using Ohm's Law.