Assertion–Reason (electrical power and heating): Assertion (A) Power loss in a conductor is P = I^2 R. Reason (R) With current density J under an applied electric field E, the heat developed per unit volume per second equals J·E.

Introduction / Context:
Joule heating describes electrical power converted to heat in conductors and resistive media. The macroscopic formula P = I^2 R is widely used, while the microscopic expression for volumetric heating q̇_v = J·E connects material properties and field distributions to thermal power density.

Given Data / Assumptions:

• Ohmic conductor with J = σE (linear isotropic medium).
• Steady sinusoidal or DC conditions (instantaneous relations apply to both).
• Uniform cross-section A and length ℓ for simple derivation.

Concept / Approach:

The local power density is q̇_v = J·E. Integrating over the conductor volume gives total power. Using Ohm’s law relationships (V = IR, I = JA, E = V/ℓ), we can reduce the volume integral to the lumped formula P = I^2 R, demonstrating that the microscopic statement explains the macroscopic result.

Step-by-Step Solution:

Verification / Alternative check:

The same conclusion follows from P = V^2/R by substituting I = V/R, consistent with energy conservation and circuit theory.

Why Other Options Are Wrong:

• If R were false, the derivation from fields to circuit formula would fail, but it is standard.

Common Pitfalls:

• Confusing magnitude J·E with vector product sign conventions; in passive media J is parallel to E, so J·E ≥ 0.

Final Answer:

Both A and R are true and R is correct explanation of A