Electromagnetics — assertion–reason on electron acceleration and current density Assertion (A): An electron of charge −e and mass m placed in a uniform electric field E experiences an acceleration a = −(e/m) * E. Reason (R): In a material containing n electrons per m^3, if each electron has drift velocity v_x along x, the current density is J = − n * e * v_x.

Introduction / Context:This item checks two cornerstone relations in electromagnetics and solid-state physics: Newton’s second law for a point charge in an electric field, and the drift-current density in a conductor or semiconductor. Distinguishing truth value from causal explanation is the key.

Given Data / Assumptions:

• Electron charge q = −e and mass m.
• Uniform electric field vector E.
• Number density of mobile electrons n (m^−3) and drift velocity component v_x.
• Classical, low-field drift (Ohmic regime), neglecting collisions during the instant of acceleration law statement.

Concept / Approach:For a point charge in an electric field, the electric force is F = q * E. Using Newton’s second law, a = F / m = (q/m) * E. For an electron (q = −e), a = −(e/m) * E, which is the assertion. For a carrier ensemble, current density by definition is charge per unit volume times average velocity: J = ρ_v * v. With electron charge −e and number density n, ρ_v = −n e, so J = −n e v; along x, J_x = −n e v_x. This is the reason statement.

Step-by-Step Solution:Compute force: F = q * E = −e * E.Apply Newton’s law: a = F/m = −(e/m) * E → A is true.Form current density: J = (charge density) * (drift velocity) = (−n e) * v → R is true.Check explanation linkage: R describes macroscopic current, not why a = −(e/m) * E; it does not causally explain A.

Verification / Alternative check:In steady Ohmic conduction, collisions limit average acceleration, giving v_d = μ E with mobility μ; nevertheless, the instantaneous acceleration law and the macroscopic J formula remain consistent.

Why Other Options Are Wrong:(a) implies R explains A, but J = −n e v does not yield a = −(e/m) * E. (c) and (d) contradict well-established relations. (e) is false because both statements are correct.

Common Pitfalls:

• Confusing drift velocity with instantaneous acceleration.
• Dropping the negative sign for electron charge in J.

Final Answer:Both A and R are true but R is not correct explanation of A