Match List-I with List-II (Symbols have their usual meanings)List-ILi…

The correct answer is Option (1) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

All four expressions come from the drift of free electrons: in a field \( E \) each electron gains the average velocity \( -eE\tau/m \) between collisions, where \( \tau \) is the relaxation time.

(A) Current density — this matches (IV) \( nev_d \). The current through area \( A \) is \( I = neAv_d \), so the current per unit area is \( j = nev_d \).

(B) Drift velocity — this matches (I) \( \displaystyle\frac{-eE\tau}{m} \). The acceleration \( -eE/m \) acts for an average time \( \tau \); the minus sign shows that electrons drift opposite to the field.

(C) Conductivity — this matches (II) \( \displaystyle\frac{ne^{2}\tau}{m} \). Putting the drift velocity into \( j = ne|v_d| \) gives \( j = (ne^{2}\tau/m)E \), and the factor multiplying \( E \) is the conductivity \( \sigma \).

(D) Mobility — this matches (III) \( \displaystyle\frac{e\tau}{m} \). Mobility is the drift speed per unit field, \( |v_d|/E = e\tau/m \).

NCERT (2026-27):
Drift of Electrons and the Origin of Resistivity – “… Thus, averaging Eq. (3.16) over the N-electrons at any given time t gives us for the average velocity v_d

    \( \displaystyle \mathbf{v}_d \equiv (\mathbf{v}_i)_{\text{average}} = -\frac{e\mathbf{E}}{m}\tau \qquad (3.17) \)

… This is the phenomenon of drift and the velocity v_d in Eq. (3.17) is called the drift velocity.”

Drift of Electrons and the Origin of Resistivity – “… Comparison with Eq. (3.13) shows that Eq. (3.22) is exactly the Ohm’s law, if we identify the conductivity \( \sigma \) as

    \( \displaystyle \sigma = \frac{n e^2 \tau}{m} \qquad (3.23) \)”

Mobility – “… An important quantity is the mobility \( \mu \) defined as the magnitude of the drift velocity per unit electric field: … Hence,

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