Match the LIST-I with LIST-II (Symbols have their usual meaning)
| LIST-I (Physical Quantity) | LIST-II (Expression) | ||
|---|---|---|---|
| A. | Drift velocity of electrons in a conductor | I. | \( -\left(\displaystyle\frac{e}{m}\right)\vec{E} \) |
| B. | Current density \( (\vec{J}) \) | II. | \( \displaystyle\frac{m}{ne^{2}\tau} \) |
| C. | Resistivity | III. | \( \mu\vec{E} \) |
| D. | Acceleration of an electron in an electric field | IV. | \( \sigma\vec{E} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → A-III, B-IV, C-II, D-I
All four expressions come from the free-electron model: in a field \( \vec{E} \) each electron is accelerated between collisions, and \( \tau \) is the average time between collisions.
(A) Drift velocity of electrons in a conductor — this matches (III) \( \mu\vec{E} \). The drift velocity is \( \vec{v}_d = -(e\tau/m)\vec{E} \), and the mobility \( \mu = e\tau/m \) is defined as the drift speed per unit field, so \( v_d = \mu E \). The mobility is a positive quantity, so this relation gives the size of the drift velocity; electrons drift opposite to \( \vec{E} \).
(B) Current density \( (\vec{J}) \) — this matches (IV) \( \sigma\vec{E} \). This is Ohm’s law in vector form.
(C) Resistivity — this matches (II) \( \displaystyle\frac{m}{ne^{2}\tau} \). Putting the drift velocity into \( J = nev_d \):
\( \displaystyle J = ne\left(\frac{e\tau}{m}E\right) = \frac{ne^{2}\tau}{m}E \)
\( \displaystyle \sigma = \frac{ne^{2}\tau}{m}, \quad \rho = \frac{1}{\sigma} = \frac{m}{ne^{2}\tau} \)
(D) Acceleration of an electron in an electric field — this matches (I) \( -\left(\displaystyle\frac{e}{m}\right)\vec{E} \). The force on the electron is \( -e\vec{E} \), so \( \vec{a} = -e\vec{E}/m \).
NCERT (2026-27):
Drift of Electrons and the Origin of Resistivity – “… Consider now the situation when an electric field is present. Electrons will be accelerated due to this field by
\( \displaystyle \mathbf{a} = -\frac{e\mathbf{E}}{m} \qquad (3.15) \)
where –e is the charge and m is the mass of an electron.”
Ohm’s Law – “… The current density, (which we have defined as the current through unit area normal to the current) is also directed along E, and is also a vector j (≡ j E/E). Thus, the last equation can be written as,
\( \displaystyle \mathbf{E} = \mathbf{j}\rho \qquad (3.12) \)
or,
\( \displaystyle \mathbf{j} = \sigma\mathbf{E} \qquad (3.13) \)
where \( \sigma \equiv 1/\rho \) is called the conductivity.”
Mobility – “… An important quantity is the mobility \( \mu \) defined as the magnitude of the drift velocity per unit electric field:
\( \displaystyle \mu = \frac{|v_d|}{E} \qquad (3.24) \)”
Temperature Dependence of Resistivity – “… From this equation, resistivity of a material is given by
\( \displaystyle \rho = \frac{1}{\sigma} = \frac{m}{n e^{2}\tau} \qquad (3.27) \)”