The electric field inside a metal wire is doubled. The temperature of the wire stays the same. How many of the following quantities will change?
Drift speed of the electrons, relaxation time, mobility of the electrons, current density, number of free electrons per unit volume, acceleration of an electron between collisions, resistivity of the metal
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Three
At a fixed temperature, the relaxation time \( \tau \) and the number density \( n \) of a metal do not depend on the field \( E \). A quantity changes only if \( E \) appears in its formula.
The quantities that change:
Drift speed, \( v_d=eE\tau/m \) — changes
Current density, \( j=\sigma E \) — changes
Acceleration between collisions, \( a=eE/m \) — changes
The quantities that do not change:
Relaxation time — depends on collisions only
Mobility, \( \mu=e\tau/m \) — has no \( E \) in it
Number of free electrons per unit volume — fixed for the metal
Resistivity, \( \rho=m/(ne^{2}\tau) \) — has no \( E \) in it
Number of quantities that change = 3
NCERT (2026-27):
Drift of Electrons and the Origin of Resistivity – “… We have, of course, made assumptions that τ and n are constants, independent of E.”
Summary – “… If such collisions occur on an average at a time interval τ, vd = aτ = eEτ/m where a is the acceleration of the electron.”
Mobility – “An important quantity is the mobility μ defined as the magnitude of the drift velocity per unit electric field: … Hence,”
\( \displaystyle \mu=\frac{v_d}{E}=\frac{e\tau}{m} \)\( \qquad (3.25) \)