Ohm’s law for metallic conduction (current density vs. electric field) Which relation correctly expresses Ohm’s law in differential form for a homogeneous metal?

Introduction / Context:
Ohm’s law at the continuum level relates current density J to electric field E via the material’s conductivity σ. This microscopic (point-form) statement underpins resistor design, field analysis, and interpretation of bulk measurements where R = ρ L/A is derived from σ = 1/ρ.

Given Data / Assumptions:

• Homogeneous, isotropic metal.
• Steady-state conditions; linear response (no significant heating nonlinearity).
• Definitions: σ is conductivity, ρ = 1/σ is resistivity.

Concept / Approach:

In differential form: J = σ E. Integrating over cross-sectional area and length recovers the familiar V = I R with R = ρ L/A. The law is empirical but arises from drift of conduction electrons with average drift velocity v_d and scattering time τ, yielding σ = n e^2 τ / m in the Drude model.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: [σ] = S/m so [J] = S/m * V/m = A/m^2, consistent.

Why Other Options Are Wrong:

(b) uses resistivity ρ; the correct relation with ρ is J = E/ρ. (c) and (d) are vague proportionalities, not equations. (e) is dimensionally wrong.

Common Pitfalls:

Mixing up σ and ρ or forgetting vector nature (J and E are generally colinear in isotropic media).

Final Answer:

J = σ E