(a) Define the term ‘conductivity’ of a metallic wire. Write its SI unit. (b) Using the concept of a free electron in a conductor, derive the expression for the conductivity of wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field E.

(a) The term 'conductivity' refers to the ability of a material, specifically a metallic wire in this context, to conduct electric current. It is a measure of how easily electric charges (typically electrons) can flow through the material when an electric field is applied. Conductivity is denoted by the symbol σ (sigma).

The SI unit of conductivity is the siemens per meter (S/m).

(b) To derive the expression for conductivity (σ) in terms of number density (n) and relaxation time (τ), we'll consider the concept of a free electron gas in a conductor. In this model, we assume that in a metal, there are a large number of electrons that are relatively free to move within the lattice structure of the material.

Let's start by considering the drift of electrons in the presence of an applied electric field (E). The equation for the average drift velocity (v_d) of electrons is given by:

v_d = μ * E

where μ (mu) is the electron mobility, a material-specific constant that relates the drift velocity to the applied electric field.

Next, we'll express the current density (J) in terms of the average drift velocity and the number density (n) of electrons:

J = n * e * v_d

where e is the charge of an electron (1.6 x 10^-19 C).

Now, we'll substitute the expression for drift velocity (v_d) in terms of electron mobility (μ) and electric field (E):

J = n * e * μ * E

Finally, we can define the conductivity (σ) as the ratio of current density (J) to the applied electric field (E):

σ = J / E

Substituting the expression for current density (J) in terms of number density (n) and electron mobility (μ), we get:

σ = (n * e * μ * E) / E

Now, the electric field (E) cancels out, and we obtain the expression for conductivity (σ) in terms of number density (n) and electron mobility (μ):

σ = n * e * μ

The relation between current density (J) and the applied electric field (E) is given by:

J = σ * E

where σ is the conductivity of the wire, and E is the applied electric field. This equation represents Ohm's law, which states that the current density is directly proportional to the electric field strength in a conductor, with the conductivity acting as the constant of proportionality.