Equation of motion for an electron in a uniform electric field in a metal A metal has a large number of free electrons. Let e be the elementary charge and m the electron mass. When a uniform electric field E is applied along the x-direction, the electrons acquire a velocity with an x-component v_x. Which relation correctly states the instantaneous equation of motion (before including scattering averages)?

Introduction / Context:
The Drude model starts from Newton’s second law applied to electrons in an applied electric field. Because electrons carry negative charge (−e), the force and acceleration directions are opposite to the electric field. Establishing the correct sign is crucial to derive drift velocity, mobility, and conductivity expressions.

Given Data / Assumptions:

• Electron charge q_e = − e with e > 0 by convention.
• Uniform, static electric field E along +x.
• Instantaneous motion before averaging over scattering time τ.

Concept / Approach:

Lorentz force (electric part) on an electron is F = q_e E = (− e) E. Newton’s law gives m dv/dt = F, hence m dv_x/dt = − e E along x. After including collisions, the average steady-state drift velocity is v_d = − (e τ / m) E, leading to current density J = n e v_d (with the sign handled conventionally) and conductivity σ = n e^2 τ / m.

Step-by-Step Solution:

Verification / Alternative check:

Direction check: E along +x pushes electrons toward −x (negative acceleration), consistent with the negative sign.

Why Other Options Are Wrong:

(b) and (c) have wrong sign; (d) gives the magnitude but wrong sign for negative carriers; (e) ignores the negative charge in the momentum growth.

Common Pitfalls:

Dropping the sign of the electron charge or confusing electron drift with conventional current direction.

Final Answer:

m dv_x/dt = − e E