Electron drift in metals — dependence on electric field In a metal under an applied electric field E (Ohmic region), the average electron drift velocity v_d is:

Introduction / Context:
In conductors obeying Ohm’s law, the current density is proportional to the electric field. On the microscopic level, this translates to a proportionality between the average drift velocity of charge carriers and the applied field.

Given Data / Assumptions:

• Metallic conductor in the linear (Ohmic) regime.
• Uniform field E, steady-state drift established.
• Temperature and microstructure fixed so mobility is constant.

Concept / Approach:
The drift velocity is defined as v_d = μ E, where μ is the carrier mobility (assumed constant in the linear regime). Since μ does not depend on E in this region, v_d ∝ E. This is consistent with J = n q v_d = σ E, where σ = n q μ is the electrical conductivity.

Step-by-Step Solution:
Start from J = σ E and J = n q v_d.Equate: n q v_d = σ E.With σ = n q μ, obtain v_d = μ E.Thus, v_d is directly proportional to E in the Ohmic range.

Verification / Alternative check:
Measured current–voltage curves of metals at moderate fields are linear; microscopic interpretation yields v_d ∝ E.

Why Other Options Are Wrong:
Inverse or quadratic dependencies occur in non-Ohmic regimes (e.g., high-field effects) and are not applicable here; “independent of E” is contrary to Ohm’s law.

Common Pitfalls:
Mixing random thermal velocity (large, field-independent) with the much smaller drift velocity (field-dependent); confusing mobility changes at high temperature with field dependence.

Final Answer:
proportional to E