A hypothetical metal is known to have an electrical resistivity of \(3.3 \times 10^{-8}(\Omega \cdot \mathrm{m}) .\) A current of \(25 \mathrm{~A}\) is passed through a specimen of this metal \(15 \mathrm{~mm}\) thick. When a magnetic field of \(0.95\) tesla is simultaneously imposed in a direction perpendicular to that of the current, a Hall voltage of \(-2.4 \times 10^{-7} \mathrm{~V}\) is measured. Compute the following: (a) the electron mobility for this metal (b) the number of free electrons per cubic meter

When we talk about electrical resistivity, we are referring to how strongly a material opposes the flow of electric current. It's a fundamental property that tells us how much resistance an object will provide to the flow of charge. The lower the resistivity, the better the material conducts electricity.

In our exercise, the resistivity of the hypothetical metal was given as \(3.3 \times 10^{-8} (\Omega \cdot \mathrm{m})\). This value allows us to infer how easily electrons move through the metal. For instance, elements like copper or silver, which are known for their excellent conductivity, have low resistivity values. On the other hand, materials with high resistivity, like rubber, act as insulators.

The relationship between electrical resistivity, electron mobility, and free electron density is given by the equation \(\rho = \frac{m}{ne\mu}\). In this context, \(m\) refers to the electron mass, \(n\) the free electron density, \(e\) the charge of an electron, and \(\mu\) the mobility of electrons. This formula was crucial in determining the number of free electrons per cubic meter for the metal in the exercise.

Electron mobility, denoted as \(\mu\), measures how fast an electron can move through a conductor when an electric field is applied. It's influenced by factors like the structure of the material, temperature, and the presence of impurities. High mobility means that the electrons can move more easily, which usually translates into good conductivity.

In the given problem, the electron mobility is calculated from the Hall coefficient \(R_H\), which is derived from the Hall effect. This effect is observed when a magnetic field is applied at a right angle to the current in a conductor, resulting in a measurable transverse voltage – the Hall voltage. The mobility is inversely related to the Hall coefficient and the charge of an electron, as shown in the calculation \(\mu = -\frac{1}{R_H e}\). High electron mobility is beneficial in electronics as it enhances the speed at which devices can operate.

Free electron density refers to the number of electrons that are available to conduct electricity in a given volume of material. When we say 'free', we mean that these electrons are not bound to atoms and can move freely within the metal. This concept is central to understanding and calculating the electrical properties of conductors.

In the exercise we explored, determining the free electron density \(n\) was a crucial step in understanding the metal's electrical characteristics. The formula that connects resistivity \(\rho\), electron mobility \(\mu\), and free electron density is rearranged to find the density: \(n = \frac{m}{\rho e\mu}\). The more free electrons per cubic meter, the greater the metal's conductivity. By calculating this density, students can get a clear picture of how well the hypothetical metal will conduct electricity under certain conditions.