Introduction:
The drift velocity of free electrons in a conductor refers to the average velocity at which electrons move in the presence of an electric field. This velocity is influenced by various factors, including the relaxation time of the electrons within the conductor. The relaxation time represents the average time it takes for an electron to collide with the atoms of the conductor.

Expression for drift velocity:
The expression for drift velocity (vd) in terms of relaxation time (τ) can be derived using the following steps:

1. Derive the equation of motion:
- Consider a conductor with length L and cross-sectional area A.
- Assume the electric field across the conductor is E.
- The force acting on an electron due to the electric field is given by F = eE, where e is the charge of an electron.
- According to Newton's second law, the force acting on an electron is equal to its mass (m) multiplied by its acceleration (a).
- Therefore, eE = ma

2. Derive the equation for acceleration:
- The acceleration of an electron can be expressed as a = (Δv/Δt), where Δv is the change in velocity and Δt is the change in time.
- Since the electrons experience numerous collisions with atoms in the conductor, their velocity does not change linearly with time.
- However, on average, the change in velocity can be expressed as Δv = (vd - 0) = vd, where vd is the drift velocity.
- Similarly, the change in time is equal to the relaxation time, Δt = τ.

3. Substitute the derived equations:
- Substituting the values from step 2 into the equation from step 1, we get eE = m(vd/τ).
- Rearranging the equation, we can solve for the drift velocity, vd = (eEτ/m).

Explanation:
The derived expression for drift velocity in terms of relaxation time indicates that the drift velocity is directly proportional to the electric field (E) and relaxation time (τ) and inversely proportional to the mass of the electrons (m). This equation shows that a stronger electric field or a longer relaxation time will result in a higher drift velocity.

The relaxation time represents the average time between collisions for electrons within the conductor. When an electric field is applied, the electrons experience a force that accelerates them. However, due to the collisions with atoms, their velocity does not increase linearly with time. Instead, the electrons reach an average drift velocity determined by the electric field strength and relaxation time.

In summary, the expression for drift velocity in terms of relaxation time provides a quantitative understanding of how free electrons move in a conductor under the influence of an electric field.