Current density for electron drift in a metal A specimen of metal contains n conduction (valence) electrons per m^3. If v_x is the average drift velocity of these electrons along x and each electron carries charge e, what is the current density J (x-component) in the metal?

Introduction / Context:
In conductors, conventional current density J points in the direction of positive-charge flow. Electrons carry negative charge, so their drift direction is opposite to J. Getting the sign right is important in microscopic transport equations and when relating J to drift velocity in the Drude model.

Given Data / Assumptions:

• Electron number density n (per m^3).
• Average electron drift velocity component v_x (along x).
• Electron charge magnitude e > 0; electron charge is −e.

Concept / Approach:

By definition, J = charge density in motion × velocity. For electrons: charge density ρ_e = (−e) n. Hence, J_x = ρ_e v_x = (−e n) v_x. If v_x is positive (electrons moving +x), J_x is negative (conventional current in −x). Many circuit texts quote the magnitude J = n e |v_d| and treat direction separately, but the vector equation includes the minus sign.

Step-by-Step Solution:

Verification / Alternative check:

Combine with Drude result v_d = (− e τ / m) E to obtain J = n e^2 τ / m E = σ E, consistent with Ohm’s law and sign conventions.

Why Other Options Are Wrong:

(a) omits the negative sign; (c), (d), (e) are dimensionally incorrect or unrelated.

Common Pitfalls:

Forgetting that v_x refers to electron motion, not conventional current direction; always include the sign associated with the carrier charge.

Final Answer:

J = − n e v_x