Current continuity in conductors: When a steady current flows through a conductor with a non-uniform cross-sectional area, how do current and current density vary along different cross-sections?

Introduction / Context:
In electrical engineering and physics, understanding how electric current behaves in conductors with changing geometry is fundamental. This question tests current continuity and the relationship between current (I), current density (J), and cross-sectional area (A) for a steady (time-invariant) current in a single connected conductor.

Given Data / Assumptions:

• Direct current or steady-state current (no charge accumulation with time).
• Conductor is continuous, with no branches and no charge sinks/sources along its length.
• Area of cross-section varies along the length (non-uniform A).

Concept / Approach:

Conservation of charge (continuity) requires the same current to flow through every cross-section in steady state. However, current density is defined as J = I / A in a uniform material when current distributes evenly across the section. If A changes, J must adjust inversely to maintain the same I. Hence, I is constant, while J varies with A.

Step-by-Step Solution:

Verification / Alternative check:

Use the integral form of the continuity equation for steady state: ∇·J = 0. For a prismatic path with varying A(s), integrating over a pillbox aligned with the conductor gives equal inflow and outflow currents, confirming I is constant while J redistributes with A.

Why Other Options Are Wrong:

• Option (a): Different I at different sections violates charge conservation in steady state.
• Option (b): Implies constant I but says nothing about J; incomplete and not the best answer to the combined assertion.
• Option (c): Suggests constant J but varying I; physically inconsistent for a single connected path.
• Option (e): Both I and J cannot remain the same when A changes; J must vary as 1/A.

Common Pitfalls:

• Confusing current with current density; current is global through the section, current density is local and area-dependent.
• Assuming branching or storage; neither applies in a single, steady path.

Final Answer:

current will be the same but current density will be different at different cross-sections.