How many of the following quantities for a metal depend on the relaxa…

The correct answer is Option (2) → Four

The relaxation time \( \tau \) is the average time between two collisions of a free electron with the ions of the metal.

The quantities that depend on τ:

• Drift speed in a given field — \( v_d=eE\tau/m \)

• Mobility — \( \mu=e\tau/m \)

• Conductivity — \( \sigma=ne^{2}\tau/m \)

• Resistivity — \( \rho=m/ne^{2}\tau \)

The quantities that do not:

• Number density of free electrons — the number of free electrons per unit volume of the metal; it is not set by collisions

• Charge of an electron — a fixed constant

• Mass of an electron — a fixed constant

Number of quantities that depend on τ = 4

NCERT (2026-27):
Summary – “… If such collisions occur on an average at a time interval τ, v_d = aτ = eEτ/m”

Mobility – “An important quantity is the mobility μ defined as the magnitude of the drift velocity per unit electric field…”

    \( \displaystyle \mu=\frac{v_d}{E}= \)\( \displaystyle \frac{e\tau}{m} \)\( \qquad (3.25) \)

Drift of Electrons and the Origin of Resistivity – “… if we identify the conductivity σ as”

    \( \displaystyle \sigma=\frac{ne^{2}}{m}\tau \)\( \qquad (3.23) \)

Temperature Dependence of Resistivity – “… ρ thus depends inversely both on the number n of free electrons per unit volume and on the average time τ between collisions. …”